Attitude Representation and Transformation Matrices

1. Attitude Representation and
Transformation Matrices
Mangal Kothari
Department of Aerospace Engineering
Indian Institute of Technology Kanpur
Kanpur - 208016

3. Coordinated Frames
• Describe relative position and orientation of objects
– Aircraft relative to direction of wind
– Camera relative to aircraft
– Aircraft relative to inertial frame
• Some things most easily calculated or described in
certain reference frames
– Newton’s law
– Aircraft attitude
– Aerodynamic forces/torques
– Accelerometers, rate gyros
– GPS
– Mission requirements

4. Rotation of Reference Frame

5. Rotation of Reference Frame

6. Particle/Rigid Body Rotation
• One can say then that a Rigid Body is essentially a Reference Frame (RF).
The translation of the origin of the RF describes the translational position. The
specific orientation of the axes wrt to a chosen Inertial Reference provides the
angular position.
Reference
Body
}
ˆ
{
Reference
Inertial
}
ˆ
{


b
n
[C] – Direction Cosine Matrix

7. Euler Angles
• Need way to describe attitude of aircraft
• Common approach: Euler angles
• Pro: Intuitive
• Con: Mathematical singularity
– Quaternions are alternative for overcoming singularity

8. Vehicle-1 Frame

9. Vehicle-2 Frame

10. Body Frame

11. Inertial Frame to Body Frame
Transformation

12. Rotational Kinematics
Inverting gives

13. Differentiation of a Vector

14. Let {b} have an angular velocity w and be expressed as:
Then
Thus
skew-symmetric
cross product
operator
But LHS
Finally
Poisson Kinematic
Equation
Nine parameter attitude
representation

15. For the Euler 3-2-1 Sequence
Attitude Kinematics Differential Equation

16. Euler’s Principal Rotation Theorem
Informal Statement: There exists a principal axis about which a single axis
rotation through F will orient the Inertial axes with the Body axes.

17. Rotational Dynamics
Newton’s 2nd Law:
• is the angular momentum vector
• is the sum of all external moments
• Time derivative taken wrt inertial frame
Expressed in the body frame,

18. Rotational Dynamics
Because is unchanging in the body frame, and
Rearranging we get
where

19. Inertia matrix

20. Rotational Dynamics
If the aircraft is symmetric about the plane, then and
This symmetry assumption helps simplify the analysis. The inverse of
becomes

21. Rotational Dynamics
Define
’s are functions of moments and products of inertia

22. Equations of Motion
gravitational, aerodynamic, propulsion
External Forces and Moments

23. Gravity Force
expressed in vehicle frame
expressed in body frame