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Attitude Representation and Transformation Matrices 3.pdf · Attitude Representation and...
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Attitude Representation and
Transformation Matrices
Mangal Kothari
Department of Aerospace Engineering
Indian Institute of Technology Kanpur
Kanpur - 208016
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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
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Rotation of Reference Frame
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Rotation of Reference Frame
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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.
ReferenceBody }ˆ{
Reference Inertial}ˆ{
b
n
[C] – Direction Cosine Matrix
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Euler Angles
• Need way to describe attitude of aircraft
• Common approach: Euler angles
• Pro: Intuitive
• Con: Mathematical singularity
– Quaternions are alternative for overcoming singularity
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Vehicle-1 Frame
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Vehicle-2 Frame
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Body Frame
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Inertial Frame to Body Frame
Transformation
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Rotational Kinematics
Inverting gives
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Differentiation of a Vector
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Let {b} have an angular velocity w and be expressed as:
Then
Thus
skew-symmetric
cross product
operator
But LHS
Finally
Poisson Kinematic
EquationNine parameter attitude
representation
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For the Euler 3-2-1 Sequence
Attitude Kinematics Differential Equation
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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.
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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,
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Rotational Dynamics
Because is unchanging in the body frame, and
Rearranging we get
where
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Inertia matrix
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Rotational Dynamics
If the aircraft is symmetric about the plane, then and
This symmetry assumption helps simplify the analysis. The inverse of
becomes
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Rotational Dynamics
Define
’s are functions of moments and products of inertia
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Equations of Motion
gravitational, aerodynamic, propulsion
External Forces and Moments
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Gravity Force
expressed in vehicle frame
expressed in body frame