**Body Fixed From Inertial:**

Flight Dynamics is the science behind the attitude or orientations of the aircraft in three dimensions. Rolling, Pitching and Yawing, Angles of rotation about the center of mass in 3D are considered three important parameters of Flight Dynamics.

**Pitching: **The Rotation of an aircraft about the lateral axis is called Pitching. And the angle between the element of a pitch cone and its axis is called Pitch Angle.

**Rolling: **The angular motion of an aircraft tending to set up a rotation about a longitudinal axis is called Rolling. And the angle between the roll cone and its axis is called roll angle.

**Yawing: **Angular Rotation of an aircraft about a vertical axis is called Yawing. The Yaw angle is measured between the relative wind and the axis of aircraft.

**Reference Frames:**

To describe the state of the aircraft, we can consider aircraft body and earth as non-inertial & inertial frames. For small unmanned aircraft, we can ignore earth rotation and curvature effects. Hence it is quite enough to use North-East-down Cartesian coordinate frame centered as the launch position of an inertial frame. Here Earth fixed axis considered as X_{E}, Y_{E}, Z_{E }and vectors along this I, J, K and Body axis considered as X_{B}, Y_{B}, Z_{B }and vectors along this i, j, k.

**Transformation:**

The Transformation to the body fixed reference frame from the earth reference frame is done by three rotations through the angle ψ, θ, ф. These angles are called Euler angles. The three rotations are not commutative, so they must be made in specified order to get desired aircraft orientation.

**Assuming:**

Pitch Euler Angle: θ

Roll Euler Angle: ф

Yaw Euler Angle: ψ

**Condition 1:**

If we rotate I, J, K about the k axis through the yaw or heading angle ψ, this produce intermediate axis i_{1}, j_{1}, k_{1.}

Hence

**Condition 2:**

Now, if we rotate i_{1}, j_{1}, k_{1 }about the j1 axis through the pitch angle θ, this produce intermediate axis i_{2}, j_{2}, k_{2:}

**Condition 3:**

Now, if we rotate i_{2}, j_{2}, k_{2 }about the i_{2} axis through the roll angle ф, this produce body axis i, j, k_{:}

Now to perform the entire transformation to the body fixed coordinate system from the inertial (earth) coordinate system, we multiply the all rotation matrices to get desired transformation matrix. So finally:

Where c, s belongs to cos and sin.

Because they are orthogonal so each matrix mentioned above has property like: Transpose of the matrix is equal to the inverse of matrix.

If we want to get I, J, K in terms of i, j, k then:

**Body Fixed From Stability:**

**Stability:**Stability coordinate system is a rotating reference frame, which is originated at the aircraft center of gravity. This is made up of the unit Vectors i_{s}, j_{s}, k_{s}and forming an orthogonal triad by the right hand rule. The X_{s}and Y_{s }form the plane of motion of the air-vehicle and Y_{s }axis aligned with Y_{B}axis. The body axes are related to the stability axes through the single rotation, defined as angle of attack denoted as α about the Ys axis.

**2. Transformation: **

The Transformation from the stability frame to the body fixed reference frame includes the single rotation about the Y_{s }axis through the angle α. It can be mentioned mathematically as:

Where *X _{b: }Body Coordinate System *

*Xs: Stability Coordinate System*

If we want to get *Xs in terms of **X _{b }*then:

**Wind from Stability:**

The wind coordinate system is a rotating reference frame also originated at center of gravity of the air vehicle. Its coordinated system axes denoted as X**Wind:**_{w}, Y_{w}, Z_{w}, vectors along this I_{w}, J_{w}, K_{w }form this right hand orthogonal system. X_{w}and Y_{w }are located in the Vehicle instantaneous plane of motion. X_{w }always aligned with the Air vehicle flight path vector relative to the surrounding air mass. Wind axes are related with stability axes through a single rotation, denoted as side slip angle β about the Z_{s }axis.

2. **Transformation:**

The Transformation to the wind reference frame from the stability reference frame includes single rotation through the side slip angle β about the Z_{s }axis.

Where *Xw _{: }Wind Coordinate System *

*Xs: Stability Coordinate System*

If we want to get *Xs in terms of **Xw *then:

**Wind from Body-Fixed:**

The Transformation to the wind coordinates from the body fixed system includes two rotations α and β, already discussed above.

T^{wb }= T^{ws }T^{sb}

Where T^{wb }: Transformation body to wind, T^{ws } : Transformation stability to wind, T^{sb}: Transformation body fixed to wind stability

To transform back in to body coordinates:

**Wind from inertial:**

All the transformation between each coordinate systems have been developed. Now the most general form of transformation inertial coordinate system to wind coordinate system can be built. From above equations:

T^{ wi } = T ^{wb }T^{bi}

Where T^{bi}: Transformation inertial to body:

And the transformation to inertial frame from wind system:

T^{iw } =

**Flight Path:**

The flight path coordinate system is a rotation reference frame, originated at center of gravity. The unit vectors along this are I_{FP}, j_{FP}, k_{FP }form right handed orthogonal system. The x axis is aligned with the flight path vector relative to inertial frame, y axis perpendicular to center of gravity vector and z axis situated between the vertical plane, which is formed by the flight path and gravity vectors. The coordinate system is defined by two rotations through the horizontal and vertical flight path angles, σ and γ. Under Zero wind condition, flight path vector relative to inertial frame is aligned with the flight path vector relative to air mass. Apart of it, air vehicle oriented with both the role angle, ф, and the angle of side slip, β, at zero. The flight path coordinate system is perfectly aligned with wind and stability coordinate system. Under these conditions γ = θ – α and σ = ψ.

**Transformation: (Flight Path from Inertial)**

The transformation from the inertial reference frame to the flight frame include two rotations, first one about the Z_{E }axis and second one about Y_{E }axis.

Now Transforming to Inertial from Flight Path: