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Department of Aerospace Engineering
IIT Kanpur, India
Flight Dynamics: Mathematical Modeling
Dr. Mangal Kothari
Department of Aerospace Engineering
Indian Institute of Technology Kanpur
Kanpur - 208016
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Particle/Rigid Body Kinematics
• Start with a choice of a coordinate reference frame wherein
the motion of the body is described (Inertial Reference)
• Kinematics relates the rate of change to “position” to
“velocities”.
• Note: The position and velocity coordinates are
dependent on the choice of the reference frame
• For particle kinematics, we are only interested in the
translational motion
• Rigid bodies – In addition to translational position, we are
also interested in the orientation/attitude or “angular
position” of the body. In many applications such as a
spacecraft pointing, we desire the spacecraft observe
something of interest that in turn requires the spacecraft to
maintain a specific orientation.
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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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4
Rotation Frame
ReferenceBody ˆ
Reference Inertialˆ
b
n
[R] – Direction Cosine Matrix
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Attitude Representation
• 3 Degrees of Freedom
• The most general representation is Rotation
matrices (SO(3))
– 9 elements
– cumbersome to use
– No singularity
• Most commonly used representation: Euler angles
– Intuitive Physical interpretation
– Minimalistic representation : 3 parameters for 3 DOF
– But exhibit a phenomenon known as Gimbal Lock
• Other parameterization
– Quaternion
– Classical and modified Rodrigues parameters
– Axis angle representation
Rotation of Reference Frame
Rotation of Reference Frame
Euler Angles
• Need way to describe attitude of aircraft
• Common approach: Euler angles
• Pro: Intuitive
• Con: Mathematical singularity
– Quaternions are alternative for overcoming singularity
Vehicle-1 Frame
Vehicle-2 Frame
Body Frame
Inertial Frame to Body Frame Transformation
Translational Kinematics
Rotational Kinematics
Inverting gives
State Equations
Six of the 12 state equations for the UAV come from the kinematic
equations relating positions and velocities:
The remaining six equations will come from applying Newton’s 2nd law
to the translational and rotational motion of the aircraft.
Attitude Representation
What does it imply mathematically?
But Euler angles do
not form an
orthogonal vector.
The Euler rates are
also not orthogonal.
At pitch 900 the
matrix becomes
singular.
Consider a 3-2-1 rotation
Note: The singularity occurs in all Euler angle rotation sequences for the middle rotation
Quaternions
• 4 component extended complex number
• Consists of scalar and vector part
• These are mathematical objects. Can also be used to represent rotations.
- Recall: what does multiplying any complex number by 𝑒𝑖θ does? It rotates the vector by θ!
• Remove singularity at the cost of one more parameter. The main reason they started being used for satellites. Now used extensively for small Aerial vehicles, aerospace robotics, VTOLs, etc.
• Simpler to compose
• Some denote it as (w,x,y,z) with w being the scalar part.
Representation No. of Parameters
Rotation Matrix 9
Euler Angles 3
Quaternions 4
Quaternion Algebra
Have their own definition of operations
Illustration with a right hand rule:
Can be used to define
quaternion product
Properties similar to complex numbers
• Norm:
• Conjugate:
• Inverse:
• Product is non-commutative:
• Product is associative:
Recall
Rotation using Quaternions
• Exercise:
Unit modulus quaternions:
Axis-angle representation
Euler's Rotation Theorem
Any rotation or sequence of rotations of a rigid body or coordinate
system about a fixed point is equivalent to a single rotation by a given
angle θ about a fixed axis (called Euler axis) that runs through the
fixed point.
• Rotation operator:
4 parameters to represent 3 degrees of freedom Must satisfy a constraint
Conventions• In loose terms, Rotation is a directional and relative quantity with a magnitude
(but remember rotation is not a vector!)• Need to first set the rules: Left or right handed?
Rotating frames (passive) or rotating vectors (active)?Direction of operation (in passive case)
Conventions
• Hamilton Notation Order: if go towards left: local to global
• Implications
Local perturbations are compounded to right (post-multiplied)
Global perturbations are compounded to left (pre-multiplied)
• Suppose 1st rotation is given by and 2nd by
if 2nd rotation is defined relatively:
if 2nd rotation is defined globally:
• Similar to Rotation Matrices! - Recall: for a 321 rotation sequence, rotation matrix for
conversion from local to Earth frame is:
What if rotations were defined always with respect to original axis?
(check for yourself!)
X
Z
Y
X
Z
Y
X
’
Y
’
Z
’
X
Z
Z
’’
Y
’’X
’’
X
Z
Y
X
’’
Z
’’
Y
’’
Y1st
rotation
2nd rotation
Case a
Case b
2nd rotation
Relative
Rotation
Global
Rotation
Derivative
By First Principles
Equivalent
formulas
If the change from
previous attitude to current
attitude is defined locally,
the change in attitude
is post-multiplied.
For small angles
Quaternions vs Euler Angles
• No singularity vs Gimbal lock
• Computationally less expensive: no trigonometric function evaluation
• No discontinuity in representation like Euler angles
Less intuitive
Dual Covering
Unit modulus constraint
Rotation matrix
Rigid Body Dynamics
Conversion between Quaternions and Euler angles
Quaternions to Euler angles:
Exercise: Try yourself!
Euler angles to Quaternions:
Quaternions corresponding
to the three rotations are
given by
Since the rotations are
relative, we post-multiply
the rotations.
For a 1-2-3 Euler
rotation:
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Rotational Kinematics
Let b have an angular velocity w and be expressed as:
Then
Thus
skew-symmetric cross
product operator
But LHS
Poisson Kinematic EquationNine parameter attitude representation
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Differentiation of a Vector
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Translational Dynamics
can be expressed in body frame as
where
Since we have that
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Rotational Dynamics
Because is unchanging in the body frame, and
Rearranging we get
where
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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
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Equation of Motion Summary
System of 12 first-order ODE’s
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Blade Element Theory
Divided rotor blade to several independent 2D airfoil elements
Side view
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Rotor Aerodynamics
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Thrust and Torque Coefficient
Integrating dCT to calculate thrust coefficient of entire rotor
Integrating dCQ to calculate torque coefficient of entire rotor
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Momentum Theory
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Momentum conservation
Energy conservation
Mass conservation
After carrying out necessary algebra, the induced inflow is given as
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Rotor Dynamics
𝑇 = 𝐶𝑇 𝜌 𝐴 𝑅2 Ω2
𝑇 = 𝐶𝑇 𝜌 𝐴 𝑅2 Ω2
Q =1
2𝐶𝑇
3
2 𝜌 𝐴 𝑅3 Ω2
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Control Mapping
45/3
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Longitudinal Aerodynamics
Longitudinal Forces – Body Frame
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Lateral Aerodynamics
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Propeller Torque
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Equations of Motion
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Linear State-space Models
nonlinear state equations
trim condition
Linearize around trim condition
deviation from trim
Re-writing state equation
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Longitudinal State-Space
Equations
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Jacobian Matrices
Take partial derivatives and evaluate them
at trim state and trim input
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Longitudinal State-space
Equations
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Alternative Form – Longitudinal
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Lateral State-space
Equations
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Jacobian Matrices
Take partial derivatives and evaluate them
at trim state and trim input
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Lateral State-space
Equations
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Alternative Form - Lateral
Questions ???