Coordinate Systems Transformation : U.A.V

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 i₁, j₁, k_1.

Hence

Condition 2:

Now, if we rotate i₁, j₁, k₁ about the j1 axis through the pitch angle θ, this produce intermediate axis i₂, j₂, k_2:

Condition 3:

Now, if we rotate i₂, j₂, k₂ about the i₂ 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:

1.  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:

1. Wind: 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_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: