aero3560a1

8/4/2019 AERO3560A1

1/16

AERO3560

FLIGHT MECHANICS 1TUTORIAL ASSIGNMENT 1

NICHOLAS GIANNELIS

308150856Monday 22nd August 2011

8/4/2019 AERO3560A1

2/16

2

Contents

Page No.

1. Question 1 31.1 Part A 3

1.2 Part B 3

2. Question 2 4

3. Question 3 5

3.1 Part A 5

3.2 Part B 6

3.3 Part C 6

4. Question 4 7

4.1 Part A 7

4.2 Part B 8

5. Appendix A-Matlab Script to Question 3 11

5.1 Part A 11

5.2 Part B 12

6. Appendix B-Matlab Script to Question 4 13

6.1 Quaternion Integration 13

6.2 Euler Angle Integration 14

8/4/2019 AERO3560A1

3/16

3

1. Question 1

1.1 Part A

J = A2500 EIn order to convert from body frame to LVLH we require the transformation matrix H.

H = H#{H${H%{{H#{I# = A1 0 00 cos I# sinI#0 s i n I# cosI#E

H${I$ = AcosI$ 0 s i n I$0 1 0sinI$ 0 cos I$ E

H%{I% = A cosI% sinI% 0sinI% cosI% 00 0 1EAs we only have a rolling body rate we can expressI#as a function of time, explicitly

I# = 25 = 5

36

The transformation matrix from the LVLH frame to the body frame thus becomes:

H = H# 536 F H${0H%{0 =

1 0 00 cos 536 5360 s i n 536 cos 536

However, we require the transformation matrix from body to LVLH; this is merely the transpose of

the above:

H =1 0 00 c o s 536 sin 5360 s i n 536 cos 536

1.2 Part BIn order to allow for simultaneous pitching the governing equations incorporate highly coupled

nonlinear terms. In order to determine the aircraft attitude one would have to calculate the Euler

angle rates of change and integrate these rates over the time of simulation. The integration

procedure may involve simple Euler integration, incrementing the original value of each Euler angle

by the product of the time differential and the rate of change.

8/4/2019 AERO3560A1

4/16

4

2. Question 2

In order to establish a transformation matrix from the Sydney reference frame (=151, = -34) to

the Saskatoon reference frame (=-106.66, = 52.13) we consider three separate rotations. The firstof which is a -124 rotation about the y-axis (easterly direction) to bring the system to the North

Pole. The transformation matrix is as below:

H${124 = Acos124 0 sin1240 1 0sin 124 0 cos124 E = A0.559 0 0.8290 1 00.829 0 0.559E

The North Pole acts as a singularity on the Earths surface with respect to the North East Down

reference frame, as such we may then make a rotation about the x-axis of -257.66 to bring the

system to the same longitude as Saskatoon as below:

H#{257 = A1 0 00 cos 257.66 sin 257.660 sin257.66 cos257.66E = A1 0 00 0.214 0.9770 0.977 0.214EThe system is then brought to the correct latitude as Saskatoon by considering a 37.87 rotation

about the y-axis:

H${37.87 = Acos37.87 0 sin37.870 1 0sin37.87 0 cos37.87 E = A0.789 0 0.6140 1 00.614 0 0.789 E

The transformation matrix relating the Sydney and Saskatoon reference frame is thus a product of

the aforementioned rotation matrices, explicitly:

H = H${124 H#{257 H${37.87The matrix multiplication is performed in Matlab, yielding:

H = A0.5502 0.8099 0.20340.5997 0.2137 0.77120.5811 0.5463 0.6033EIn order to best describe the relative orientation of a person standing at Saskatoon with respect to

Sydney reference frame we consider the Saskatoon up vector. We take the Inverse of the Htransformation matrix and multiply this by a unit up vector in the Saskatoon frame, yielding:H# = H = A0.5502 0.5997 0.58110.8099 0.2137 0.54630.2034 0.7712 0.6033 E

A0.5502 0.5997 0.58110.8099 0.2137 0.54630.2034 0.7712 0.6033 E A001E = A

0.54630.58110.6033EIn the Sydney reference frame. The Saskatoon up vector is illustrated in dark blue below:

8/4/2019 AERO3560A1

5/16

5

-0.5

0

0.5

1

-0.1

00.1

0.2

0.3

0.40.5

0.6

0.7

0.8

0.9

-1

-0.8

-0.6

-0.4

-0.2

0

North

Sydney North-East-Down

East

Down

Saskatoon Up Vector

North

East

Down

3. Question 3

3.1Part AThe Matlab script written to convert Euler Angles to Quaternions is provided in the appendix. The

table of provided Euler Angles is supplied below, followed by the corresponding Quaternions output

from the function file E2Q.

EulerAngles

Case

1 0 0 90

2 0 -90 0

3 -90 0 0

4 0 90 -90

5 45 45 456 45 -30 100

7 -150 -10 -120

8 97 -103 43

Quaternions

Case " # $ %1 0.7071 0 0 0.7071

2 0.7071 0 -0.7071 0

3 0.7071 -0.7071 0 0

4 0.5000 0.5000 0.5000 -0.50005 0.8446 0.1913 0.4619 0.1913

8/4/2019 AERO3560A1

6/16

6

6 0.4977 0.4208 0.1295 0.7473

7 0.0560 -0.5007 0.8221 -0.2654

8 0.1690 0.6239 -0.3116 0.6965

3.2Part BThe Matlab script to convert from Quaternions to Euler Angles is provided in the appendix. The table

below provides the Quaternions, their conversion to Euler angles and a conversion back to

Quaternions using the E2Q function to confirm its functionality.

Quaternions

Case " # $ %1 0.5862 -0.6941 0.1349 0.3954

2 0.4822 -0.8698 0.0914 -0.0507

Euler AnglesCase

1 -90.0046 45.0007 23.0034

2 -121.9938 -0.0030 -11.9992

E2Q(ans)

Case " # $ %1 0.5862 -0.6941 0.1349 0.3954

2 0.4822 -0.8698 0.0914 -0.0507

As the conversion confirms the precise input arguments we may ascertain the codes function

correctly.

3.3Part C>> Q2E(ans)

Case

1 0 0 90.0000

2 0 -90.0000 0

3 -90.0000 0 0

4 0 90.0000 -90.00005 45.0000 45.0000 45.0000

6 45.0000 -30.0000 100.0000

7 -150.0000 -10.0000 -120.0000

8 -83.0000 -77.0000 -137.0000

In employing the inversion quaternion function to convert back to Euler angles it is evident that all

bar the last case are identical to the input arguments. Although the final arguments may appear to

differ when input into the trigonometric functions governing the quaternion and Euler angle

conversion equations they yield identical results. As such the script cannot discern betweentrigonometrically identical arguments.

8/4/2019 AERO3560A1

7/16

7

0 2 4 6 8 10 12 14 16 18 20

-200

-150

-100

-50

0

50

100

150

200Euler Angles Derived from Quaternion Integration

Time(s)

g

(

g)

Phi

Theta

Psi

0 2 4 6 8 10 12 14 16 18 20

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1Quaternions Derived from Quaternion Integration

Time(s)

Quaternions(rad)

e0

e1

e2

e3

0 2 4 6 8 10 12 14 16 18 200

50

100

150

200

250

300Euler Angles Derived Directly from Euler Rates

Time(s)

EulerAngles(deg)

Phi

Theta

Psi

0 2 4 6 8 10 12 14 16 18 20-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Quaternions Derived Directly from Euler Rates

Time(s)

Quaternions(rad)

e0

e1

e2

e3

4. Question 4

4.1Part APresented below are the output plots of Euler angle and Quaternion variation with time, as derivedfrom the quat_int and euler_int functions over a twenty second time interval with yaw rate of 5.6

seconds (Student Number 308150856).

Quaternion Integration with normalising correction

Euler angle Direct Integration

8/4/2019 AERO3560A1

8/16

8

0 2 4 6 8 10 12 14 16 18 20-200

-150

-100

-50

0

50

100

150

200

250


Time(s)

EulerAngles(d

eg)

Phi

Theta

Psi

0 2 4 6 8 10 12 14 16 18 20-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1Quaternions Derived from Quaternion Integration

Time(s)

Quaternions(ra

d)

e0

e1

e2

e3

Discussion of difference in results

As evident when comparing the prior Euler angle and Quaternion integration not a great deal of

discrepancy can be seen, bar a distinct jump in the Quaternion integration Euler Angle plot. The

Quaternion angles are bounded between the range 180 and thus the apparent discontinuity is

simply a switch from 180 to -180, the same point. The Euler angles derived from the Euler rate

integration however are unbounded and continuous.

In order to better observe any difference in results the plots for Euler rate and Quaternion rate

integration are overlayed below with dashed segments representing Euler rate integration and solid

lines Quaternion integration. Small differences are evident as the time of simulation increase, with

the curves appearing to diverge. This divergence is an expected characteristic as the direct

integration of Euler rates accumulates the error resulting from the linear segmented Euler

integration method. The Quaternion integration however employs a normalising factor in order to

mitigate this error at each time step. As such as the length of simulation is increased we would

expect a greater divergence of the two curves. Further the timestep of 0.1 s is quite large relative to

the rotation rates and as such has quite a bearing on the results. The Quaternions split this time step

across four variables rather than three, again mitigating error.

4.2Part BBy imposing a zero roll rate on the system it necessarily assures that the aircraft will exhibit a vertical

nose up attitude, the point of singularity regarding Euler angle attitude representation. Distinct

discrepancies are evident between the quaternion and Euler rate integration methods with zero roll

rate. It is evident that at the point of singularity (approximately 15 seconds) theta attains a

maximum value regarding quaternion integration and phi and psi attain their equivalent

trigonometric values. The Euler rate integration plot however permits the unbounded increase of

theta.

8/4/2019 AERO3560A1

9/16

9

0 2 4 6 8 10 12 14 16 18 20-100

-50

0

50

100

150


Time(s)

EulerAngles(deg)

Phi

Theta

Psi

0 2 4 6 8 10 12 14 16 18 20-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8Quaternions Derived from Quaternion Integration

Time(s)

Quaternions(rad)

e0

e1

e2

e3

0 2 4 6 8 10 12 14 16 18 20-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8Quaternions Derived Directly from Euler Rates

Time(s)

Quaternions(rad)

e0

e1

e2

e3

0 2 4 6 8 10 12 14 16 18 200

20

40

60

80

100

120


Time(s)

EulerAngles(deg)

Phi

Theta

Psi

r = 0 deg/s

8/4/2019 AERO3560A1

10/16

10

0 2 4 6 8 10 12 14 16 18 20-100

-50

0

50

100

150


Time(s)

EulerAng

les(deg)

Phi

Theta

Psi

0 2 4 6 8 10 12 14 16 18 20-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8Quaternions Derived from Quaternion Integration

Time(s)

Quaternions(rad)

e0

e1

e2

e3

0 2 4 6 8 10 12 14 16 18 200

200

400

600

800

1000

1200


Time(s)

Phi

Theta

Psi

0 2 4 6 8 10 12 14 16 18 20-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8Quaternions Derived Directly from Euler Rates

Time(s)

Quaternions(rad)

e0

e1

e2

e3

As another means of comparison we consider the system experiencing a near zero roll rate. At the

singularity point it is evident the Euler rate integration yields nonsensical results, indicating an

increase in phi and psi on the order of 1200%. The Quaternion integration procedure remains well

behaved at this point of singularity. The Quaternion plots accentuate this interpretation, as we see

kinks in the Quaternion plot at the singularity point whereas the Quaternion rate integration

maintains a smooth continuous curve.

8/4/2019 AERO3560A1

11/16

11

5. Appendix A-Matlab Script Q3

5.1 Part A-Euler Angle to Quaternions

function [ Quaternions ] = E2Q( EulerAngles )%Function file to convert Euler angles to Quaternions% Each set of Euler angles is entered as a row vector% EulerAngles = [phi1 theta1 psi1;phi2 theta2 psi2;etc]% Quaternions are output in sets of row vectors corresponding to% the relevant Euler angle row vector% Quaternions = [e01 e11 e21 e31;e02 e12 e22 e32;etc]

%%%Convert Euler angles to half angles in radianseuler = 0.5*EulerAngles*pi/180;

%%%Convert Euler angle to Quaternion components through relevant

% equations (Lecture Notes Week 2 Pg 37a)e0 = cos(euler(:,3)).*cos(euler(:,2)).*sin(euler(:,1))-sin(euler(:,3))...

.*sin(euler(:,2)).*cos(euler(:,1));

e1 =cos(euler(:,3)).*cos(euler(:,2)).*cos(euler(:,1))+sin(euler(:,3))...

.*sin(euler(:,2)).*sin(euler(:,1));

e2 =cos(euler(:,3)).*sin(euler(:,2)).*cos(euler(:,1))+sin(euler(:,3))...

.*cos(euler(:,2)).*sin(euler(:,1));

e3 = -cos(euler(:,3)).*sin(euler(:,2)).*sin(euler(:,1))+sin(euler(:,3))...

.*cos(euler(:,2)).*cos(euler(:,1));

%%%Check root sum square of quaternion components to ensure%%%equations have been input in correct formatmusquared = e0.^2+e1.^2+e2.^2+e3.^2;

if musquared > 2fprintf('Calculation Error-Root Sum square too large\n');

end

%%%Output Quaternion MatrixQuaternions = [e0 e1 e2 e3];

end

M-file to call Euler angle conversion

clcclear allclose allEulerAngles = [0 0 90;0 -90 0;-90 0 0;0 90 -90; 45 45 45;45 -30

100;-150 -10 -120;97 -103 43]E2Q(EulerAngles)

8/4/2019 AERO3560A1

12/16

12

5.2 Part B-Quaternions to Euler Anglesfunction [ EulerAngles ] = Q2E( quat )%Function file to convert Quaternions to Euler angles% Each set of Quaternions is entered as a row vector% quat = [e01 e11 e21 e31;e02 e12 e22 e32;etc]

% Euler Angles are output in sets of row vectors corresponding to% the relevant Quaternion row vector% EulerAngles = [phi1 theta1 psi1;phi2 theta2 psi2;etc]

%%%Convert Quaternions to Euler Angles through relevant equations% (Lecture Notes Week 2 Pg 37)phi =atan2((quat(:,3).*quat(:,4)+quat(:,1).*quat(:,2)),(quat(:,1)...

.^2+quat(:,4).^2-0.5));

theta = atan2((quat(:,1).*quat(:,3)-quat(:,2).*quat(:,4)),((quat(:,1)...

.^2+quat(:,2).^2-0.5).^2+(quat(:,2).*quat(:,3)+quat(:,1).*quat(:,4)).^2).^0.5);

psi =atan2((quat(:,2).*quat(:,3)+quat(:,1).*quat(:,4)),(quat(:,1)...

.^2+quat(:,2).^2-0.5));

%%%Output Euler Angles in degreesEulerAngles = [phi theta psi]*180/pi;

end

M-file to call Quaternion conversion

clcclear allclose allquat = [0.5862 -0.6941 0.1349 0.3954;0.4822 -0.8698 0.0914 -0.0507]Q2E(quat)

8/4/2019 AERO3560A1

13/16

13

6. Appendix B-Matlab Script Q4

6.1 Quaternion Integration

function [ euler,quat ] = quat_int( initial_euler, body_rates )%Function file integrating Quaternion rates to determine aircraft attitude% Initial conditions are input as below% initial_euler = [phi theta psi]-initial Euler angles% body_rates = [p q r]-initial rotation rates about body axis

% Both Euler angles and Quaternions are output as matricies with a unit% timestep increase from row to row.% Plots are output for both Quaternion and Euler angle variation with% time

%%%Initialise Euler angles vector for initial valuesphi = initial_euler(1);

theta = initial_euler(2);

psi = initial_euler(3);

euler = [phi theta psi];

%%%Convert Euler angles to Quaternions for Quaternion Integration, take% transpose of vector for subsequent calculations such that values% increasing with timestep are stored as new columns in the Quaternion% (quat) matrixquat = E2Q(euler)';

%%%Initialise body rotation rates in rad/secp = body_rates(1)*pi/180;

q = body_rates(2)*pi/180;

r = body_rates(3)*pi/180;

%%%Initialise Time/Step Variablestime = 20; %Duration of simulation

dt = 0.1; %Time step

n = time/dt; %Number of iterations

%%%Input equation for Quaternion rates calculation as per Lecture Notes% Week 2 Pg 37.quat_dot(:,:) = 0.5*[-quat(2,:) -quat(3,:) -quat(4,:);...

quat(1,:) -quat(4,:) quat(3,:);...quat(4,:) quat(1,:) -quat(2,:);...-quat(3,:) quat(2,:) quat(1,:)]*[p;q;r];

%%%Define for loop for duration of simulation, apply Euler Integrationfor i = 1:n

%%%Euler Integration

quat(:,i+1) = quat(:,i)+quat_dot(:,i)*dt;

8/4/2019 AERO3560A1

14/16

14

%%%Normalise Quaternions for correctionMu = (quat(1,i+1)^2+quat(2,i+1)^2+quat(3,i+1)^2+quat(4,i+1)^2)^0.5;

quat(:,i+1) = quat(:,i+1)./Mu;

%%%Increment Euler angle ratesquat_dot(:,i+1) = 0.5*[-quat(2,i) -quat(3,i) -quat(4,i);...

quat(1,i) -quat(4,i) quat(3,i);...quat(4,i) quat(1,i) -quat(2,i);...-quat(3,i) quat(2,i) quat(1,i)]*[p;q;r];

end

%%%Transpose quaternion vector to output as matrix with unit time step% increasing from row to rowquat = quat';

%%%Convert quaternions to Euler angleseuler = Q2E(quat);

%%%Plot Resultsplot(0:0.1:20, euler);title('Euler Angles Derived from Quaternion Integration');xlabel('Time(s)');ylabel('Euler Angles (deg)');legend('Phi','Theta','Psi');grid on;grid minor;

figure

plot(0:0.1:20, quat);title('Quaternions Derived from Quaternion Integration');xlabel('Time(s)');ylabel('Quaternions (rad)');legend('e0','e1','e2','e3');grid on;grid minor;end

M-file to call Quaternion Integration

clc;clear all;

close all;initial_euler = [0 0 90];body_rates = [0 6 5.6];quat_int(initial_euler,body_rates)

6.2 Euler Angle Integrationfunction [ euler,quaternions ] = euler_int( initial_euler, body_rates )%Function file integrating Euler rates to determine aircraft attitude% Initial conditions are input as below% initial_euler = [phi theta psi]-initial Euler angles% body_rates = [p q r]-initial rotation rates about body axis

% Both Euler angles and Quaternions are output as matricies with a unit% timestep increase from row to row.

8/4/2019 AERO3560A1

15/16

15

% Plots are output for both Quaternion and Euler angle variation with% time

%%%Initialise euler angles in radiansphi = initial_euler(1)*pi/180;

theta = initial_euler(2)*pi/180;

psi = initial_euler(3)*pi/180;

euler = [phi; theta; psi];

%%%Initialise body rotation rates in rad/secp = body_rates(1)*pi/180;

q = body_rates(2)*pi/180;

r = body_rates(3)*pi/180;

%%%Initialise Time/Step Variablestime = 20; %Duration of simulation

dt = 0.1; %Time step

n = time/dt; %Number of iterations

%%%Input equation for Euler rates calculation as per Lecture Notes Week 2% Pg 31euler_dot(:,:) = [1 sin(euler(1,:))*tan(euler(2,:))cos(euler(1,:))*tan(euler(2,:)); ...

0 cos(euler(1,:)) -sin(euler(1,:));...0 sin(euler(1,:))*sec(euler(2,:)) cos(euler(1,:))*sec(euler(2,:))]*...[p;q;r];

%%%Define for loop for duration of simulation, apply Euler Integration

for i = 1:n%%%Euler Integrationeuler(:,i+1) = euler(:,i)+euler_dot(:,i)*dt;

%%%Increment Euler angle rateseuler_dot(:,i+1) = [1 sin(euler(1,i))*tan(euler(2,i))

cos(euler(1,i))*tan(euler(2,i)); ...

0 cos(euler(1,i)) -sin(euler(1,i));...0 sin(euler(1,i))*sec(euler(2,i)) cos(euler(1,i))*sec(euler(2,i))]*...[p;q;r];

end

%%%Transpose euler vector to output as matrix with unit time step% increasing from row to roweuler = (euler*180/pi)';

%%%Convert Euler angles to quaternionsquaternions = E2Q(euler)

%%%Plot Resultsplot(0:0.1:20, euler);title('Euler Angles Derived Directly from Euler Rates');

8/4/2019 AERO3560A1

16/16

16

xlabel('Time(s)');ylabel('Euler Angles (deg)');legend('Phi','Theta','Psi');grid on;grid minor;

figure

plot(0:0.1:20, quaternions);title('Quaternions Derived Directly from Euler Rates');xlabel('Time(s)');ylabel('Quaternions (rad)');legend('e0','e1','e2','e3');grid on;grid minor;end

M-file to call Euler Angle Integration

clc;clear all;close all;initial_euler = [0 0 90];body_rates = [0 6 5.6];euler_int(initial_euler,body_rates)

aero3560a1

Documents