Aircraft Flight Path & Attitude

– Basic Aircraft Kinematics

– Translational Motion Model

– Concept of Inertial Attitude

– Euler Angles, Euler Rates & Transformations

Basic Aircraft Kinematics

In the study of aircraft flight dynamics, our primary

interest is the determination of vehicle response which is

captured through a set of applicable motion variables.

This requirement brings into picture, the basic discipline

of kinematics, which deals with the geometry of

translation and rotation of moving bodies.

In general, the position of aircraft can be measured with

respect to a basic inertial reference frame having three

mutually perpendicular directions.

Similarly, attitude of aircraft can be measured using

three mutually independent rotations with respect to an

inertial frame translated to vehicle centre of mass.

Translational Position

In aircraft flight dynamics, translational (or rectilinear)

position of the centre of mass is portrayed by the three

components x, y, z, representing the northerly, easterly and

vertical displacements from the origin.

By convention, ‘z’ is positive downwards so that

coordinate axes follow the right hand rule. Further,

height ‘h’ of aircraft above ground is considered to be

positive so that h = -z.

Further, the position vector ‘r’, measured from origin, is

expressed as follows.

{ } { } { }1 2 3{ }

T T Tr r r r x y z x y h−�≜ ≜ ≜

Translational Position & Velocity

The aircraft’s ‘distance’ from origin can be defined as

follows.

Next, the aircraft’s inertial velocity vector can be defined

as follows.

1/2

2 2 2 { } { }T T

x

r r x y z x y z y r r

z

= + + =

� � �≜ ≜

1 2 3

2 2 2 1/2

1 2 3

{ } { } { }

{ }

T T

I

T

I I I I

v r v v v x y z

V v v v v v v v

= = =

= = = + + =

� �ɺ ɺ ɺ ɺ≜

� � �

Translational Velocity

Cartesian description of velocities is more conveniently

represented in polar coordinate system, as shown below.

1 2 3{ } { } { cos cos cos sin sin }T T

Iv v v v V V Vγ ξ γ ξ γ= −�≜

Angular Orientation & Rates

Similar to translational motion parameters, angular

orientation and rates are defined with respect to an

inertial frame of reference at the aircraft centre of mass.

In aircraft flight dynamics context, the orientation with

respect to an inertial reference frame is described using

three Euler angles, which are measured in degrees or

radians and are respectively; ψψψψ (yaw angle), θθθθ (pitch angle)

and φφφφ (roll angle).

Euler angles represent an ordered set of sequential

rotations from an inertial reference to a reference frame

fixed to the body. The ordering is arbitrary, but must be

retained once it is chosen.

Angular Orientation

Consider the following the angular orientation.

Here, {x’, y’, z’} is inertial and {xB, yB, zB} is body frame.

Angular Orientation

In case of right handed systems, there are a total twelve

possible rotational sequences, among which the

conventional sequence is as follows,

A right-handed rotation ‘ψψψψ’ about z’ axis, followed by

another right-handed pitch rotation (θθθθ) about an inertial

axis that is coincident with the intermediate span wise (yB)

body axis; followed by one more right-handed roll

rotation (φ) about an inertial axis that is coincident with

the body’s intermediate centre line (xB).

This is also popularly called the 3-2-1 sequence,

corresponding to notation that 1 is ‘φφφφ’, 2 is ‘θθθθ’ and 3 is ‘ψψψψ’.

Inertial & body axes are coincident if all angles are zero.

Angular Orientation

The sequence 3-2-1 is graphically shown below.

Angular Orientation

The three Euler angles are independent of each other and

they can be combined to give aircraft attitude vector, as

follows.

Many times, we need to resolve the inertial velocity

vector into the body frame and Euler angles are essential

to such a resolution. Consider the body frame velocity

vector as defined below.

Here, H represents an orthonormal transformation

matrix for projecting a vector from inertial frame into the

body frame, while preserving the vector’s magnitude.

{ } { }Tφ θ ψΘ

�≜

{ } { }T B

B I Iv u v w H v=�� �

≜

Angular Orientation

The transformation matrix H can be obtained in terms of

the Euler angles as follows.

We can obtain the expressions for u, v & w using this

matrix, through matrix multiplication operation. E.g.

component ‘u’ can be obtained as follows.

Similarly, expressions for v & w can also be obtained.

cos cos cos sin sin

cos sin sin sin cos cos cos sin sin sin sin cos

sin sin cos sin cos sin cos cos sin sin cos cos

B

IH

θ ψ θ ψ θ

φ ψ φ θ ψ φ ψ φ θ ψ φ θ

φ ψ φ θ ψ φ ψ φ θ ψ φ θ

=

− − + + + − +

�

1 2 3cos cos cos sin sinu v v vθ ψ θ ψ θ= + −

Angular Orientation

Alternatively, we can define H matrix as a successive

multiplication of component matrices defined below.

In case, we know body frame velocity and we need the

inertial frame values, we can use the following relation.

1 2

1

2 1

2 2 1

cos sin 0 cos 0 sin

sin cos 0 ; 0 1 0

0 0 1 sin 0 cos

1 0 0

0 cos sin ; ;

0 sin cos

I

B B B

I I

H H

H H H H H

ψ ψ θ θ

ψ ψ

θ θ

φ φ

φ φ

− = − =

= = −

� �

� � � � �

( ) ( )1

{ } { } { } { }T

I I I

I B B B B B Bv H v H v H v

−

= = =� � �� � � �

Angular Rates

Euler angle rate vector is simply the term-by-term

derivative of the Euler angle vector, as given below.

It should be noted that not only the components of Euler

angle rate vector are not measured along three orthogonal

directions, but also transforming it into another coordinate

system is a complicated process. Nevertheless, it is

possible to express angular rate as an orthogonal vector, as

follows.

{ }1 0 sin

{ } { }, 0 cos sin cos

0 sin cos cos

B B

B E Ep q r L L

θ

ω φ φ θ

φ φ θ

− = = Θ −

�� �� ɺ≜

{ }{ } φ θ ψΘ�ɺ ɺ ɺ ɺ≜

Angular Rates

Here, p, q, r are the rates along the orthogonal axes, while

L is the transformation which is synthesized as follows.

It is found that while, individual transformations are

orthonormal, the combination is not as shown below.

LBE is also singular with respect to θθθθ = ±90o.

( )1

1 sin tan cos tan

0 cos sin

0 sin sec cos sec

E B

B EL L

φ θ φ θ

φ φ

φ θ φ θ

− = −

� �≜

[ ] 2

2 2 1

0 0

0 0

0 0

B B

n

p

q I H H H

r

φ

θ

ψ

= + +

ɺ� � �

ɺ

ɺ

Summary

The kinematics of aircraft involves writing equation for

the translational motion of its centre of mass, along with

rotation equation for attitude determination.

It is possible to arrive at transformations that convert all

motion quantities from inertial to body frame and vice

versa.

Euler angles, used for velocity and attitude determination,

contain a singularity when pitch angle is ±90o.