Matlab Roll Angle Equation

MATLAB ROLL EQUATIONS VERIFICATION

1. Tangential Component of Gravitational Vector Equation

gt = (((Vg*g)/((Vgab)^2)) * Vg);

MATLAB coding:

clear

clc

close all

syms gt Vgab VN VE VD g Vg go

Vg = [VN VE VD];

Vmag = sqrt(VN^2+VE^2+VD^2)

g = [0;0;go];

C = Vg*g;

U = [VN VE VD]* [0; 0; VD*go]

Vmag1 = Vmag^2

gt = ((U)/(Vmag1));

Results MATLAB:

gt = (VD^2*go)/(VD^2 + VE^2 + VN^2)

2. Normal Component of Gravitacional Vector

gn = g – gt

MATLAB coding:

clear

clc

close all

syms gt Vgab VN VE VD g Vg go gn

Vg = [VN VE VD];

Vgab = sqrt(VN^2+VE^2+VD^2)

g = [0 0 go];

gt = ([0 0 VD^2*go])/(VD^2 + VE^2 + VN^2)

gn = g - gt

Results MATLAB:

gn =

[ 0, 0, go - (VD^2*go)/(VD^2 + VE^2 + VN^2)]

Further Simplification by hand

= (go(VE^2 +VN^2))/VD^2 + VE^2 + VN^2

3. Tangential Component of Acceleration Vector Equation

At = ((Vg*Ag)/Vmag^2)* Vg

MATLAB coding:

clear

clc

close all

syms AN AE AD VN VE VD Vg Vmag At

Vg = [VN VE VD];

Vmag = sqrt(VN^2+VE^2+VD^2);

Ag = [AN AE AD];

C = Vg.*Ag

K = C.*Vg

At = (K)/(sqrt(VN^2+VE^2+VD^2))^2

Results MATLAB:

At =

[ (AN*VN^2)/(VD^2 + VE^2 + VN^2), (AE*VE^2)/(VD^2 + VE^2 + VN^2), (AD*VD^2)/(VD^2 + VE^2 + VN^2)]

4. Normal Component of Acceleration Vector

An = Ag – At

MATLAB coding:

clear

clc

close all

syms AN AE AD VN VE VD Vg Vmag At

Vg = [VN VE VD];

Vmag = sqrt(VN^2+VE^2+VD^2);

Ag = [AN AE AD];

C = Vg.*Ag

K = C.*Ag

At = (K)/(sqrt(VN^2+VE^2+VD^2))^2

An = Ag - At

Results MATLAB:

An =

[ AN - (AN*VN^2)/(VD^2 + VE^2 + VN^2), AE - (AE*VE^2)/(VD^2 + VE^2 + VN^2), AD - (AD*VD^2)/(VD^2 + VE^2 + VN^2)]

Simplify by hand

[ AN(VD^2 + VE^2 )/ (VD^2 + VE^2 + VN^2), AE(VD^2+VN^2)/ (VD^2 + VE^2 + VN^2),AD(VE^2+VN^2)/ (VD^2 + VE^2 + VN^2)]

5. Lift Vector

l = An - gn

MATLAB coding:

clear

clc

close all

syms AN AE AD VN VE VD Vg Vmag At l

syms gt g go gn

g = [0 0 go];

gt = ([0 0 VD^2*go])/(VD^2 + VE^2 + VN^2)

gn = g - gt

Vg = [VN VE VD];

Vmag =...