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**Frisbee flight in Matlab, can't run (Input argument "x" is undefined)**

I'm trying to use a frisbee flight simulation made by Sarah Hummel, but I'm having a hard time running it in Matlab.

I get the first message when I press run in the program and the one in red when running in the subroutine. Any ideas on how to fix this? Thanks!

Here's the code for the program:

Code:

```
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% File: simulate_flight.m
%% By: Sarah Hummel
%% Date: July 2003
%%
%% This MATLAB program allows the simulation of a single frisbee flight
%% given initial conditions and a set of aerodynamic coefficients.
%% Calls subroutine discfltEOM.m, the equations of motion for the frisbee.
%% Inertial xyz coordinates = forward, right and down
%%
%% before executing code (as described below):
%% 1) specify value for "CoefUsed"
%% 2) specify which values for the damping coefficients, use long flight or short
%% flight values.
%% 3) specify Simulation set of initial conditions: thetao, speedo, betao, and po
%% 4) specify which "x0" command to use
%% 5) specify values for "tfinal" and "nsteps"
%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
clear
format short
global m g Ia Id A d rho
global CLo CLa CDo CDa CMo CMa CRr
global CL_data CD_data CM_data CRr_rad CRr_AdvR CRr_data
global CMq CRp CNr
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% Set "CoefUsed" = 1 OR 2
%% This chooses the values of coefficients (specifies a set of if/then statements)
%% to use for CLo CLa CDo CDa CMo CMa CRr.
%% CoefUsed = 1 ... choose for using estimated short flights lift, drag, moment coefficients
%% CoefUsed = 2 ... choose for using Potts and Crowther (2002) lift, drag, moment coefficients
CoefUsed=2;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% define non-aerodynamic parameters
m = 0.175; % Kg
g = 9.7935; % m/s^2
A = 0.057; % m^2
d = 2*sqrt(A/pi); % diameter
rho = 1.23; % Kg/m^3
Ia = 0.002352; % moment of inertia about the spinning axis
Id = 0.001219; % moment of inertia about the planar axis'
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% THE THREE ESTIMATED COEFFICIENTS
%CMq= -0.005, CRp =-0.0055, CNr = 0.0000071 % short (three) flights
CMq= -1.44E-02 ; CRp =-1.25E-02; CNr = -3.41E-05; % long flight f2302
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% THE seven COEFFICIENTS estimated from three flights
CLo= 0.3331;
CLa= 1.9124;
CDo= 0.1769;
CDa= 0.685;
CMo= -0.0821;
CMa= 0.4338;
CRr= 0.00171; % for nondimensionalization = sqrt(d/g), magnitude of CRr changes
% with nondimensionalization
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% THE seven COEFFICIENTS from Potts and Crowther (2002)
%% CL =[ rad CL deg]
CL_data=[ -0.1745 -0.2250 -10
-0.05236 0 -3
0 0.150 0
0.08727 0.4500 5
0.17453 0.7250 10
0.26180 0.9750 15
0.34907 1.2000 20
0.43633 1.4500 25
0.52360 1.6750 30];
%% CD =[ rad CD deg]
CD_data=[ -0.1745 0.1500 -10
-0.05236 0.0800 -3
0 0.1000 0
0.08727 0.1500 5
0.1745 0.2600 10
0.26180 0.3900 15
0.3491 0.5700 20
0.4363 0.7500 25
0.5236 0.9200 30];
%% CM =[ rad CM deg]
CM_data=[-0.174532925 -0.0380 -10
-0.087266463 -0.0220 -5
-0.052359878 -0.0140 -3
0 -0.0060 0
0.052359878 -0.0060 3
0.104719755 -0.0020 6
0.157079633 0.0000 9
0.20943951 0.0100 12
0.261799388 0.0220 15
0.34906585 0.0440 20
0.401425728 0.0600 23
0.453785606 0.0840 26
0.523598776 0.1100 30];
%%CRr_deg=[-5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 30 ]
CRr_rad = [-0.0873 -0.0698 -0.0524 -0.0349 -0.0175 0.0000 0.0175 0.0349 0.0524 0.0698 0.0873 0.1047 0.1222 0.1396 0.1571 0.1745 0.1920 0.2094 0.2269 0.2443 0.2618 0.5236];
CRr_AdvR= [2 1.04 0.69 0.35 0.17 0];
CRr_data =[
-0.0172 -0.0192 -0.0180 -0.0192 -0.0180 -0.0172 -0.0172 -0.0168 -0.0188 -0.0164 -0.0136
-0.0100 -0.0104 -0.0108 -0.0084 -0.0080 -0.0080 -0.0060 -0.0048 -0.0064 -0.0080 -0.0030
-0.0112 -0.0132 -0.0120 -0.0132 -0.0120 -0.0112 -0.0112 -0.0108 -0.0128 -0.0104 -0.0096
-0.0068 -0.0072 -0.0076 -0.0052 -0.0048 -0.0048 -0.0028 -0.0032 -0.0048 -0.0064 -0.003
-0.0056 -0.0064 -0.0064 -0.0068 -0.0064 -0.0064 -0.0052 -0.0064 -0.0028 -0.0028 -0.004
-0.002 -0.004 -0.002 -0.0016 0 0 0 0 -0.002 -0.0048 -0.003
-0.0012 -0.0016 -0.0004 -0.0028 -0.0016 -0.0016 -0.0004 0.0004 0.0004 0.0008 0.0004
0.0008 0.0012 0.0008 0.002 0.0028 0.0032 0.0024 0.0028 0.0004 -0.0012 -0.003
-0.0012 -0.0012 -0.0016 -0.0016 -0.0012 -0.0004 0.0004 0.0008 0.0008 0.0016 0.0004
0.0020 0.0004 0.0016 0.002 0.002 0.002 0.0012 0.0012 0 -0.0012 -0.003
-0.0012 -0.0012 -0.0004 -0.0008 -0.0008 -0.0008 0.0004 0.0004 0.0004 0.0008 0.0004
0.0008 -0.0004 0.0000 0.0000 0.0004 0.0000 0.0000 0.0004 -0.002 -0.0012 -0.003];
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% angle and angular velocities in rad and rad/sec respectively
%% phi = angle about the x axis phidot = angular velocity
%% theta = angle about the y axis thetadot = angular velocity
%% gamma = angle about the z axis gd(gammadot) = angular velocity
%% For reference, two sets of previously used initial conditions...
%% Long flight (f2302) release conditions:
%% thetao = 0.211; speedo = 13.5; betao = 0.15; gd=54
%% Common release conditions:
%% thetao = 0.192; speedo = 14; betao = 0.105; gd=50
%% Define Simulation set initial conditions, enter your choosen values here:
thetao = .192; % initial pitch angle
speedo = 13.7; % magnitude, m/sec
betao = .105; % flight path angle in radians between velocity vector and horizontal
gd=50; % initial spin
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
alphao = thetao - betao; % inital alpha
vxo = speedo * cos(betao); % velocity component in the nx direction
vzo = -(speedo * sin(betao)); % velocity component in the nz direction
% (note: nz is positive down)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% x0= vector of initial conditions
%% x0= [ x y z vx vy vz phi theta phidot thetadot gd gamma]
%% Specify the set of inital conditions to use:
%% the first set of conditions is for a disc released:
%% theta, speed, and spin as specified above (thetao, speedo, gd),
%% 1 meter above the ground, forward and right 0.001,
%% no roll angle, no velocity in the the y direction
%% and for positive alpha, disc is pitched up, with a neg. w component
%% the second set is the long flight f2302 estimated initial conditions
%% First set:
%x0= [ 0.001 0.001 -1 vxo 0 vzo 0 thetao 0.001 0.001 gd 0 ]
%% Second set:
x0=[-9.03E-01 -6.33E-01 -9.13E-01 1.34E+01 -4.11E-01 1.12E-03 -7.11E-02 2.11E-01 -1.49E+01 -1.48E+00 5.43E+01 5.03E+00];
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% Enter values for tfinal and nsteps:
tfinal = 1.46; % length of flight
nsteps = 292; % number of time steps for data
tspan=[0:tfinal/nsteps:tfinal];
options=[];
%options = odeset('AbsTol', 0.000001,'RelTol', 0.00000001,'OutputFcn','odeplot');
%% Calls the ODE and integrate the frisbee equations of motions in the
%% subroutine, discfltEOM.m
[t,x]=ode45(@discfltEOM,tspan,x0,options,CoefUsed);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% Remaining commands are associated with creating plots of the output. A "v"
%% is put at the end of the variable to differentiate it from the variable
%% calculated in discfltEOM.m
%% give states names .... v for view for making plots
vxv = x(:,4);
vyv = x(:,5);
vzv = x(:,6);
fv = x(:,7);
thv = x(:,8);
stv = sin(thv);
ctv = cos(thv);
sfv = sin(fv);
cfv = cos(fv);
fdv = x(:,9);
thdv= x(:,10);
pv = x(:,11);
velv = [vxv vyv vzv]; %expressed in N
omegaD_N_inCv = [fdv.*ctv thdv fdv.*stv + pv]; % expressed in c1,c2,c3
for i=1:size(t)
i=i;
vmagv(i) = norm(velv(i,:)); % a nx1 row vector
%% T_c_N=[ct st*sf -st*cf;
%% 0 cf sf;
%% st -ct*sf ct*cf]
T_c_N=[ ctv(i) stv(i)*sfv(i) -stv(i)*cfv(i);
0 cfv(i) sfv(i);
stv(i) -ctv(i)*sfv(i) ctv(i)*cfv(i)];
c3v(i,:)=T_c_N(3,:); % c3 expressed in N frame
vc3v(i,:)=dot(velv(i,:),c3v(i,:)); % velocity (scalar) in the c3 direction
vpv(i,:)= [velv(i,:)-vc3v(i,:).*c3v(i,:)]; % subtract c3 velocity component to get the
% velocity vector projected onto the plane
% of the disc, expressed in N
vpmagv(i) = norm(vpv(i,:));
uvpv(i,:)=vpv(i,:)/vpmagv(i);
ulatv(i,:)=cross(c3v(i,:)',uvpv(i,:)')'; % unit vector perp. to v and c3, points right
alphav(i) = atan(vc3v(i,:)/norm(vpv(i,:)));
omegaD_N_inNv(i,:) = (T_c_N'*omegaD_N_inCv(i,:)')'; % expressed in n1,n2,n3
omegavpv(i,:) = dot(omegaD_N_inNv(i,:),uvpv(i,:)); %omega about vp axis
omegalatv(i,:) = dot(omegaD_N_inNv(i,:),ulatv(i,:));
omegaspinv(i,:)= dot(omegaD_N_inNv(i,:),c3v(i,:));
end %for i=1:size(t)
Adpv = A*rho*vmagv.*vmagv/2;
wuns=ones(size(alphav));
AdvRv=d*omegaspinv/2./vmagv';
if CoefUsed==1 % using short flights coefficients
alphaeq= -CLo/CLa; % this is angle of attack at zero lift
CLv = CLo*ones(size(alphav)) + CLa*alphav;
CDv = CDo*ones(size(alphav)) + CDa*(alphav-alphaeq*ones(size(alphav))).*...
(alphav-alphaeq*ones(size(alphav)));
CMv = CMo*wuns + CMa*alphav;
CRrv=CRr;
%CRrv= CRr*d*omegaspinv/2./vmagv';
%CRrv= CRr*sqrt(d/g)*omegaspinv;
% above line produces NaN, so leave it in Mvp equation
%Mvp = Adp*d* (CRrv*d*omegaspin/2/vmag + CRp*omegavp)*uvp; % expressed in N
Mvpv = Adpv*d* (sqrt(d/g)*CRrv*omegaspinv + CRp*omegavpv)*uvpv; % Roll moment
end % if CoefUsed==1 % using short flights coefficients
if CoefUsed==2 % using Potts and Crowther (2002) coefficients
%% interpolation of Potts and Crowther (2002) data
CLv = interp1(CL_data(:,1), CL_data(:,2), alphav,'spline');
CDv = interp1(CD_data(:,1), CD_data(:,2), alphav,'spline');
CMv = interp1(CM_data(:,1), CM_data(:,2), alphav,'spline');
CRrv = interp2(CRr_rad,CRr_AdvR,CRr_data,alphav,AdvRv','spline');
Mvpv = Adpv'*d.* (CRrv' + CRp*omegavpv); % Roll moment
end % CoefUsed==2 % using potts coefficients
liftv=CLv.*Adpv;
dragv=CDv.*Adpv;
Mlatv = Adpv'*d.* (CMv' + CMq*omegalatv); % Pitch Moment
Mspinv= [CNr*(fdv.*stv +pv)] ; % Spin Down Moment
%% Plot four subplots in one figure, the force and moments of the simulations
figure(1)
clf
subplot(2,2,1),plot(t,liftv)
title('Liftv')
subplot(2,2,2),plot(t,dragv)
title('Dragv')
subplot(2,2,3),plot(t,Mvpv,t,Mlatv)
title(' Mvpv, Mlatv')
xlabel('time(sec)')
subplot(2,2,4),plot(t,Mspinv)
title(' Mspinv')
xlabel('time(sec)')
veloc=sqrt(x(:,4).^2 +x(:,5).^2 +x(:,6).^2);
mechenrgy=m*(-2*g*x(:,3) + x(:,4).^2 +x(:,5).^2 +x(:,6).^2) /2;
Ho = mechenrgy/m/g;
figure(2)
plot(t,mechenrgy,t,veloc,'.',t,Ho,t,AdvRv);
legend('mechenrgy','vel','Ho','AdvRv')
xlabel('time(sec)')
figure(3)
plot(t,alphav)
title('Angle of Attack, \alpha')
xlabel('time(sec)')
ylabel('(rad)')
grid
%% Plot four subplots in one figure, the states
figure(4)
subplot(2,2,1),plot(t,x(:,1),'k-',t,x(:,2),'k--',t,x(:,3),'k:','LineWidth',2)
set(gca,'LineWidth',1);
set(gca,'FontName','arial');
set(gca,'FontSize',11);
%xlabel('Time (sec)')
ylabel('position (m)')
legend('x','y','z');
%axis tight
%Ylim([-2 16])
grid
subplot(2,2,2),plot(t,x(:,4),'k-',t,x(:,5),'k--',t,x(:,6),'k:','LineWidth',2)
set(gca,'LineWidth',1);
set(gca,'FontName','arial');
set(gca,'FontSize',11);
%xlabel('Time (sec)')
ylabel('velocity (m/sec)')
%title('Velocity of CM')
legend('x\prime','y\prime','z\prime');
%axis tight
%Ylim([-2 14])
%set(gca,'YTickLabel',{});
grid
subplot(2,2,3),plot(t,x(:,7),'k-',t,x(:,8),'k--','LineWidth',2)
set(gca,'LineWidth',1);
set(gca,'FontName','arial');
set(gca,'FontSize',11);
xlabel('time (sec)')
ylabel('orientation (rad)')
%title('Phi, Theta')
%title('Disc plane orientation')
legend('\phi','\theta');
grid
subplot(2,2,4),plot(t,x(:,9),'k-',t,x(:,10),'k--',t,0.1*x(:,11),'k:','LineWidth',2)
set(gca,'LineWidth',1);
set(gca,'FontName','arial');
set(gca,'FontSize',11);
xlabel('time (sec)')
ylabel('angular velocity (rad/sec)')
%title('Angular velocities')
legend('\phi\prime','\theta\prime','0.1*\gamma\prime');
grid
```

Code:

```
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% File: discfltEOM.m
%% By: Sarah Hummel
%% Date: July 2003
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function xdot=discfltEOM(~,x,CoefUsed)
% Equations of Motion for the frisbee
% The inertial frame, xyz = forward, right and down
global m g Ia Id A d rho
global CLo CLa CDo CDa CMo CMa CRr
global CL_data CD_data CM_data CRr_rad CRr_AdvR CRr_data
global CMq CRp CNr
% x = [ x y z vx vy vz f th fd thd gd gamma]
% 1 2 3 4 5 6 7 8 9 10 11 12
%% give states normal names
vx = x(4);
vy = x(5);
vz = x(6);
f = x(7);
th = x(8);
st = sin(th);
ct = cos(th);
sf = sin(f);
cf = cos(f);
fd = x(9);
thd= x(10);
gd = x(11);
%% Define transformation matrix
%% [c]=[T_c_N] * [N]
T_c_N=[ct st*sf -st*cf; 0 cf sf; st -ct*sf ct*cf];
%% [d]=[T_d_N] * [N]
%T_d_N(1,:)=[cg*ct sg*cf+sf*st*cg sf*sg-st*cf*cg];
%T_d_N(2,:)=[ -sg*ct cf*cg-sf*sg*st sf*cg+sg*st*cf];
%T_d_N(3,:)=[ st -sf*ct cf*ct]
[~,eval]=eig(T_c_N);
eigM1=diag(eval);
m1=norm(eigM1(1));
m2=norm(eigM1(2));
m3=norm(eigM1(3));
c1=T_c_N(1,:); % c1 expressed in N frame
c2=T_c_N(2,:); % c2 expressed in N frame
c3=T_c_N(3,:); % c3 expressed in N frame
%% calculate aerodynamic forces and moments
%% every vector is expressed in the N frame
vel = [vx vy vz]; %expressed in N
vmag = norm(vel);
vc3=dot(vel,c3); % velocity (scalar) in the c3 direction
vp= [vel-vc3*c3]; % subtract the c3 velocity component to get the velocity vector
% projected onto the plane of the disc, expressed in N
alpha = atan(vc3/norm(vp));
Adp = A*rho*vmag*vmag/2;
uvel = vel/vmag; % unit vector in vel direction, expressed in N
uvp = vp/norm(vp); % unit vector in the projected velocity direction, expressed in N
ulat = cross(c3,uvp); % unit vec perp to v and d3 that points to right, right?
%% first calc moments in uvp (roll), ulat(pitch) directions, then express in n1,n2,n3
omegaD_N_inC = [fd*ct thd fd*st+gd]; % expressed in c1,c2,c3
omegaD_N_inN = T_c_N'*omegaD_N_inC'; % expressed in n1,n2,n3
omegavp = dot(omegaD_N_inN,uvp);
omegalat = dot(omegaD_N_inN,ulat);
omegaspin = dot(omegaD_N_inN,c3); % omegaspin = p1=fd*st+gd
AdvR= d*omegaspin/2/vmag ; % advanced ration
if CoefUsed==1 % using short flights coefficients
CL = CLo + CLa*alpha;
alphaeq = -CLo/CLa; % this is angle of attack at zero lift
CD = CDo + CDa*(alpha-alphaeq)*(alpha-alphaeq);
CM=CMo + CMa*alpha;
%CRr= CRr*d*omegaspinv/2./vmagv';
%CRr= CRr*sqrt(d/g)*omegaspinv; % this line produces NaN, so leave it in Mvp equation
%Mvp = Adp*d* (CRr*d*omegaspin/2/vmag + CRp*omegavp)*uvp; % expressed in N
Mvp = Adp*d* (sqrt(d/g)*CRr*omegaspin + CRp*omegavp)*uvp; % expressed in N
end % if CoefUsed==1 % using short flights coefficients
if CoefUsed==2 % using potts coefficients
%% interpolation of Potts and Crowther (2002) data
CL = interp1(CL_data(:,1), CL_data(:,2), alpha,'spline');
CD = interp1(CD_data(:,1), CD_data(:,2), alpha,'spline');
CM = interp1(CM_data(:,1), CM_data(:,2), alpha,'spline');
CRr = interp2(CRr_rad,CRr_AdvR,CRr_data,alpha,AdvR,'spline');
Mvp = Adp*d* (CRr* + CRp*omegavp)*uvp; % Roll moment, expressed in N
end % if CoefUsed==2 % using potts coefficients
lift = CL*Adp;
drag = CD*Adp;
ulift = -cross(uvel,ulat); % ulift always has - d3 component
udrag = -uvel;
Faero = lift*ulift + drag*udrag; % aero force in N
FgN = [ 0 0 m*g]'; % gravity force in N
F = Faero' + FgN;
Mlat = Adp*d* (CM + CMq*omegalat)*ulat; % Pitch moment expressed in N
Mspin = [0 0 +CNr*(omegaspin)]; % Spin Down moment expressed in C
M = T_c_N*Mvp' + T_c_N*Mlat' + Mspin'; % Total moment expressed in C
% set moments equal to zero if wanted...
% M=[0 0 0];
% calculate the derivatives of the states
xdot = vel';
xdot(4) = (F(1)/m); %accx
xdot(5) = (F(2)/m); %accy
xdot(6) = (F(3)/m); %accz
xdot(7) = fd;
xdot(8) = thd;
xdot(9) = (M(1) + Id*thd*fd*st - Ia*thd*(fd*st+gd) + Id*thd*fd*st)/Id/ct;
xdot(10) = (M(2) + Ia*fd*ct*(fd*st +gd) - Id*fd*fd*ct*st)/Id;
fdd=xdot(9);
xdot(11) = (M(3) - Ia*(fdd*st + thd*fd*ct))/Ia;
xdot(12) = x(11);
xdott=xdot';
% calculate angular momentum
H = [Id 0 0 ; 0 Id 0; 0 0 Ia]*omegaD_N_inC';
format long;
magH = norm(H);
format short;
state=x';
```