Rotation sequences and different MATLAB interpretation issue: MATLAB seems to interpret rotation sequences in a quite different mode with respect to scientific literature

Hi all,

I wrote some code to understand how MATLAB interprets rotations. I have an aircraft with nose pointing NORTH (x axis), right wing pointing EAST (y axis) and the bottom of the fuselage pointing DOWN (z axis).

1) How can I plot according to my coordinate system instead of MATLAB default???

2) Why does the rotation sequence 1) yaw 2) pitch 3) roll seem to be wrong into MATLAB?

I tried using different terns of angles but the final orientation of my aircraft doesn't correspond to that I suppose to be.

E.g. let's take yaw=20*pi/180 pitch=45*pi/180 and roll=0: I'm expecting my aircraft to rotate first around DOWN of 20° and then nose UP of 45° with wings lying on NORTH-EAST plane...NO!!! IT DOESN'T!!! Why?

It seems to be a strange interpretation of aircraft convention of angle rotation.

Please help me! Thanks a lot. Luca.

Here is the code:

..............................................................

clear all
close all
clc

f=linspace(-10,10,50);
a=linspace(-5,5,50);
c=linspace(-3,0,50);

fuselage=[f; zeros(1,size(f,2)); zeros(1,size(f,2))];
wings=[zeros(1,size(f,2)); a; zeros(1,size(f,2))];
tail=[min(f)*ones(1,size(f,2)); zeros(1,size(f,2)); c];

%%Coordinates definition

% Using right-hand coordinate system:
% NORTH (positive x pointing the nose of the fuselage)
% EAST(positive y through right wing)
% DOWN (positive z pointing under the fuselage)

D=[zeros(1,size(f,2)); zeros(1,size(f,2)); linspace(0,2,50)]; % Coordinate DOWN
E=[zeros(1,size(f,2)); linspace(0,7,50); zeros(1,size(f,2))]; % Coordinate EAST
N=[linspace(0,20,50); zeros(1,size(f,2)); zeros(1,size(f,2))]; % Coordinate NORTH

%%Angles definition
% Yaw is positive rotating around the DOWN axis clockwise according to right-hand rule 

% Pitch is positive rotating around the EAST axis clockwise according to right-hand
% rule 

% Roll is positive rotating around the NORTH axis clockwise according to right-hand
% rule 

yaw=45*pi/180;
pitch=45*pi/180;
roll=0*pi/180;

attitude_aircraft(D, N, E, fuselage, wings, tail, yaw, pitch,roll);

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

function attitude_aircraft(D, N, E, fuselage, wings, tail, yaw, pitch,roll)

% Rotation according to MATLAB

R = angle2dcm(yaw, pitch, roll );

% Rotation with separates matrices 

R_x = @(x) [ 1     0     0;...
             0  cos(x) sin(x);...
             0 -sin(x) cos(x)];

R_y = @(y) [cos(y) 0 -sin(y);...
               0   1    0;...
            sin(y) 0  cos(y)];

R_z = @(z) [  cos(z) sin(z) 0;...
             -sin(z) cos(z) 0;...
                 0      0   1];          

% Rotation according to the SEQUENCE1  1)Yaw, 2)Pitch, 3)Roll              

R_corr = R_x(roll) * R_y(pitch) * R_z(yaw);

% Rotation according to the SEQUENCE2 1)Roll, 2)Pitch, 3)Yaw              

R_fissa = R_z(yaw) * R_y(pitch) * R_x(roll);

% Aircraft rotated according to MATLAB

fuso=R*fuselage;
wingsi=R*wings;
taila=R*tail;

% Aircraft rotated according to SEQUENCE1

FUSO=R_corr*fuselage;
wingsI=R_corr*wings;
tailA=R_corr*tail;

% Aircraft rotated according to SEQUENCE2

FUSOo=R_fissa*fuselage;
wingsIi=R_fissa*wings;
tailAa=R_fissa*tail;

figure (1)
hold on
xlabel('asse x');
ylabel('asse y');
zlabel('asse z');
axis equal
axis vis3d
grid on
plot3(N(1,:),N(2,:),N(3,:),'.k');
h2=plot3(E(1,:),E(2,:),E(3,:),'.k');
h3=plot3(D(1,:),D(2,:),D(3,:),'.k');

plot3(fuso(1,:),fuso(2,:),fuso(3,:),'.y');
h5=plot3(wingsi(1,:),wingsi(2,:),wingsi(3,:),'.y');
h6=plot3(taila(1,:),taila(2,:),taila(3,:),'.y');

plot3(FUSO(1,:),FUSO(2,:),FUSO(3,:),'.c');
h8=plot3(wingsI(1,:),wingsI(2,:),wingsI(3,:),'.c');
h9=plot3(tailA(1,:),tailA(2,:),tailA(3,:),'.c');

plot3(FUSOo(1,:),FUSOo(2,:),FUSOo(3,:),'.r');
h11=plot3(wingsIi(1,:),wingsIi(2,:),wingsIi(3,:),'.r');
h12=plot3(tailAa(1,:),tailAa(2,:),tailAa(3,:),'.r');

set(get(get(h2,'Annotation'),'LegendInformation'),...
    'IconDisplayStyle','off');
set(get(get(h3,'Annotation'),'LegendInformation'),...
    'IconDisplayStyle','off');
set(get(get(h5,'Annotation'),'LegendInformation'),...
    'IconDisplayStyle','off');
set(get(get(h6,'Annotation'),'LegendInformation'),...
    'IconDisplayStyle','off');
set(get(get(h8,'Annotation'),'LegendInformation'),...
    'IconDisplayStyle','off');
set(get(get(h9,'Annotation'),'LegendInformation'),...
    'IconDisplayStyle','off');
set(get(get(h11,'Annotation'),'LegendInformation'),...
    'IconDisplayStyle','off');
set(get(get(h12,'Annotation'),'LegendInformation'),...
    'IconDisplayStyle','off');

legend('fixed coordinate system','angle2dcm',...
    'SEQUENCE1',...
    'SEQUENCE2')

hold off
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Rotation around DOWN(yaw) only

fuso1=R_z(yaw)*fuselage;
wings1=R_z(yaw)*wings;
tail1=R_z(yaw)*tail;

% Rotation around DOWN(yaw) then EAST(pitch)

fuso2=R_y(-pitch)*fuso1;
wings2=R_y(-pitch)*wings1;
tail2=R_y(-pitch)*tail1;

% Rotation around DOWN(yaw) then EAST(pitch) then NORTH(roll)

fuso3=R_x(-roll)*fuso2;
wings3=R_x(-roll)*wings2;
tail3=R_x(-roll)*tail2;

figure (2)
hold on
xlabel('asse x');
ylabel('asse y');
zlabel('asse z');
axis equal
axis vis3d
grid on

plot3(N(1,:),N(2,:),N(3,:),'.k');
h14=plot3(E(1,:),E(2,:),E(3,:),'.k');
h15=plot3(D(1,:),D(2,:),D(3,:),'.k');

%%Ruotato lungo z di yaw
plot3(fuso1(1,:),fuso1(2,:),fuso1(3,:),'.b');
h17=plot3(wings1(1,:),wings1(2,:),wings1(3,:),'.b');
h18=plot3(tail1(1,:),tail1(2,:),tail1(3,:),'.b');

%%Ruotato lungo z di yaw e lungo y di pitch
plot3(fuso2(1,:),fuso2(2,:),fuso2(3,:),'.g');
h20=plot3(wings2(1,:),wings2(2,:),wings2(3,:),'.g');
h21=plot3(tail2(1,:),tail2(2,:),tail2(3,:),'.g');

%%Ruotato lungo z di yaw, lungo y di pitch e infine di roll lungo x
plot3(fuso3(1,:),fuso3(2,:),fuso3(3,:),'.m');
h23=plot3(wings3(1,:),wings3(2,:),wings3(3,:),'.m');
h24=plot3(tail3(1,:),tail3(2,:),tail3(3,:),'.m');

%%Ruotato lungo z di yaw, lungo y di pitch e di roll lungo x
plot3(FUSOo(1,:),FUSOo(2,:),FUSOo(3,:),'xr');
h26=plot3(wingsIi(1,:),wingsIi(2,:),wingsIi(3,:),'xr');
h27=plot3(tailAa(1,:),tailAa(2,:),tailAa(3,:),'xr');
% mi aspetto che la x rossa coincida con . il magenta 

set(get(get(h14,'Annotation'),'LegendInformation'),...
    'IconDisplayStyle','off');
set(get(get(h15,'Annotation'),'LegendInformation'),...
    'IconDisplayStyle','off');
set(get(get(h17,'Annotation'),'LegendInformation'),...
    'IconDisplayStyle','off');
set(get(get(h18,'Annotation'),'LegendInformation'),...
    'IconDisplayStyle','off');
set(get(get(h20,'Annotation'),'LegendInformation'),...
    'IconDisplayStyle','off');
set(get(get(h21,'Annotation'),'LegendInformation'),...
    'IconDisplayStyle','off');
set(get(get(h23,'Annotation'),'LegendInformation'),...
    'IconDisplayStyle','off');
set(get(get(h24,'Annotation'),'LegendInformation'),...
    'IconDisplayStyle','off');
set(get(get(h26,'Annotation'),'LegendInformation'),...
    'IconDisplayStyle','off');
set(get(get(h27,'Annotation'),'LegendInformation'),...
    'IconDisplayStyle','off');

legend('fixed coordinate system','DOWN(yaw)',...
    'DOWN(yaw)+EAST(pitch)',...
    'DOWN(yaw)+EAST(pitch)+NORTH(roll)',...
    'SEQUENCE2')

hold off

Answer 1

[Simulink Dude]

You will have to map the meaning of your coordinate frame in the real world to that of the animation environment.

I used it in this example:

http://www.mathworks.com/matlabcentral/fileexchange/31142-international-flight-route-planning-simulator-using-aerospace-blockset