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TRANSCRIPT
Rigid Body Rotation
Speaker: Xiaolei Chen Advisor: Prof. Xiaolin Li
Department of Applied Mathematics and Statistics
Stony Brook University (SUNY)
Content
Introduction
Angular Velocity
Angular Momentum and Inertia Tensor
Eulerโs Equations of Motion
Euler Angles and Euler Parameters
Numerical Simulation
Future work
Introduction
Question: How many degrees of freedom in the rigid
body motion?
Answer: In total, there are 6 degrees of freedom.
1) 3 for the translational motion
2) 3 for the rotational motion
Translational Motion: center of mass velocity
Rotational Motion: angular velocity, euler angles, euler
parameters
Angular Velocity
In physics, the angular velocity is defined as the rate of
change of angular displacement.
In 2D cases, the angular velocity is a scalar.
Equation: ๐๐
๐๐ก= ๐ค
Linear velocity: ๐ฃ = ๐ค ร ๐
Angular Velocity
In the 3D rigid body rotation, the angular velocity is a
vector, and use ๐ค = (๐ค1, ๐ค2, ๐ค3) to denote it.
Angular Momentum
In physics, the angular momentum (๐ฟ) is a measure of the
amount of rotation an object has, taking into account its
mass, shape and speed.
The ๐ฟ of a particle about a given origin in the body axes is
given by
๐ฟ = ๐ ร ๐ = ๐ ร (๐๐ฃ)
where ๐ is the linear momentum, and ๐ฃ is the linear velocity.
Linear velocity: ๐ฃ = ๐ค ร ๐๐ฟ = ๐๐ ร ๐ค ร ๐ = ๐(๐ค๐2 โ ๐(๐ โ ๐ค))
Angular Momentum
Each component of the angular Momentum is
๐ฟ1 = ๐(๐2 โ ๐ฅ2)๐ค1 โ๐๐ฅ๐ฆ๐ค2 โ๐๐ฅ๐ง๐ค3
๐ฟ2 = โ๐๐ฆ๐ฅ๐ค1 +๐(๐2 โ ๐ฆ2)๐ค2 โ๐๐ฆ๐ง๐ค3
๐ฟ3 = โ๐๐ง๐ฅ๐ค1 โ๐๐ง๐ฆ๐ค2 +๐(๐2 โ ๐ง2)๐ค3
It can be written in the tensor formula.๐ฟ = ๐ผ๐ค
๐ผ =
๐ผ๐ฅ๐ฅ ๐ผ๐ฅ๐ฆ ๐ผ๐ฅ๐ง๐ผ๐ฆ๐ฅ ๐ผ๐ฆ๐ฆ ๐ผ๐ฆ๐ง๐ผ๐ง๐ฅ ๐ผ๐ง๐ฆ ๐ผ๐ง๐ง
, ๐ผ๐๐ = ๐(๐2๐ฟ๐๐ โ ๐๐)
Inertia Tensor
Moment of inertia is the mass property of a rigid body
that determines the torque needed for a desired
angular acceleration about an axis of rotation. It can be
defined by the Inertia Tensor (๐ผ), which consists of nine
components (3 ร 3 matrix).
In continuous cases,
๐ผ๐๐ = ๐
๐ ๐ ๐2๐ฟ๐๐ โ ๐๐ ๐๐
The inertia tensor is constant in the body axes.
Inertia Tensor
Eulerโs Equations of Motion
Apply the eigen-decomposition to the inertial tensor and
obtain
๐ผ = ๐ฮ๐๐ , ฮ = ๐๐๐๐ ๐ผ1, ๐ผ2, ๐ผ3
where ๐ผ๐ are called principle moments of inertia, and the
columns of ๐ are called principle axes. The rigid body rotates in angular velocity (๐ค๐ฅ, ๐ค๐ฆ, ๐ค๐ง) with respect to the
principle axes.
Specially, consider the cases where ๐ผ is diagonal.
๐ผ = ๐๐๐๐ ๐ผ1, ๐ผ2, ๐ผ3 , ๐ฟ๐ = ๐ผ๐๐ค๐ , ๐ = 1,2,3
The principle axes are the same as the ๐ฅ, ๐ฆ, ๐ง axes in the
body axes.
Eulerโs Equations of Motion
The time rate of change of a vector ๐บ in the space axes
can be written as๐๐บ
๐๐ก๐
=๐๐บ
๐๐ก๐
+๐ค ร ๐บ
where ๐๐บ
๐๐ก ๐is the time rate of change of the vector ๐บ in
the body axes.
Apply the formula to the angular momentum ๐ฟ๐๐ฟ
๐๐ก๐
=๐๐ฟ
๐๐ก๐
+๐ค ร ๐บ
Eulerโs Equations of Motion
Use the relationship ๐ฟ = ๐ผ๐ค to obtain๐(๐ผ๐ค)
๐๐ก+ ๐ค ร ๐ฟ =
๐๐ฟ
๐๐ก๐
= ๐
The second equality is based on the definition of torque.๐๐ฟ
๐๐ก=๐(๐ ร ๐)
๐๐ก= ๐ ร
๐๐
๐๐ก= ๐ ร ๐น = ๐
Eulerโs equation of motion
๐ผ1 ๐ค1 โ๐ค2๐ค3 ๐ผ2 โ ๐ผ3 = ๐1๐ผ2 ๐ค2 โ๐ค3๐ค1 ๐ผ3 โ ๐ผ1 = ๐2๐ผ3 ๐ค3 โ๐ค1๐ค2 ๐ผ1 โ ๐ผ2 = ๐3
Euler Angles
The Euler angles are three angles introduced to describe
the orientation of a rigid body.
Usually, use ฯ, ๐, ๐ to denote them.
The order of the rotation steps matters.
Euler Angles
Euler Angles
Three rotation matrices are
๐ท =๐๐๐ ๐ ๐ ๐๐๐ 0โ๐ ๐๐๐ ๐๐๐ ๐ 0
0 0 1
๐ถ =1 0 00 ๐๐๐ ๐ ๐ ๐๐๐0 โ๐ ๐๐๐ ๐๐๐ ๐
๐ต =๐๐๐ ๐ ๐ ๐๐๐ 0โ๐ ๐๐๐ ๐๐๐ ๐ 0
0 0 1
Then, the rotation matrix is
๐ด = ๐ต๐ถ๐ท =
๐๐๐ ๐๐๐๐ ๐ โ ๐๐๐ ๐๐ ๐๐๐๐ ๐๐๐ ๐๐๐ ๐๐ ๐๐๐ + ๐๐๐ ๐๐๐๐ ๐๐ ๐๐๐ ๐ ๐๐๐๐ ๐๐๐โ๐ ๐๐๐๐๐๐ ๐ โ ๐๐๐ ๐๐ ๐๐๐๐๐๐ ๐ โ๐ ๐๐๐๐ ๐๐๐ + ๐๐๐ ๐๐๐๐ ๐๐๐๐ ๐ ๐๐๐ ๐๐ ๐๐๐
๐ ๐๐๐๐ ๐๐๐ โ๐ ๐๐๐๐๐๐ ๐ ๐๐๐ ๐
This matrix can be used to convert the coordinates in the
space axes to the coordinates in the body axes.
The inverse is ๐ดโ1 = ๐ด๐.
Assume ๐ฅ is the coordinates in the space axes, and ๐ฅโฒ is the
coordinates in the body axes.
๐ฅโฒ = ๐ด๐ฅ๐ฅ = ๐ด๐๐ฅโฒ
Euler Angles
Euler Angles
The relationship between the angular velocity and the
euler angles is๐ค1
๐ค2
๐ค3
=00 ๐+ ๐ท๐
๐00
+ ๐ท๐๐ถ๐00 ๐= ๐ฝโ1
๐ ๐ ๐
๐ฝโ1 =0 ๐๐๐ ๐ ๐ ๐๐๐๐ ๐๐๐0 ๐ ๐๐๐ โ๐ ๐๐๐๐๐๐ ๐1 0 ๐๐๐ ๐
๐ ๐ ๐
=
โ๐ ๐๐๐๐๐๐ก๐ ๐๐๐ ๐๐๐๐ก๐ 1๐๐๐ ๐ ๐ ๐๐๐ 0
๐ ๐๐๐๐๐ ๐๐ โ๐๐๐ ๐๐๐ ๐๐ 0
๐ค1
๐ค2
๐ค3
Euler Parameters
The four parameters ๐0, ๐1, ๐2, ๐3 describe a finite about an arbitrary axis. The parameters are defined as
Def: ๐0 = ๐๐๐ ๐
2
And ๐1, ๐2, ๐3 = ๐sin(๐
2)
Euler Parameters
In this representation, ๐ is the unit normal vector, and ๐ is the rotation angle.
Constraint: ๐02 + ๐1
2 + ๐22 + ๐3
2 = 1
๐ด =
๐02 + ๐1
2 โ ๐22 โ ๐3
2 2(๐1๐2 + ๐0๐3) 2(๐1๐3 โ ๐0๐2)
2(๐1๐2 โ ๐0๐3) ๐02 โ ๐1
2 + ๐22 โ ๐3
2 2(๐2๐3 + ๐0๐1)
2(๐1๐3 + ๐0๐2) 2(๐2๐3 โ ๐0๐1) ๐02 โ ๐1
2 + ๐22 โ ๐3
2
Euler Parameters
The relationship between the angular velocity and the
euler parameters is ๐0 ๐1 ๐2 ๐3
=1
2
โ๐1 โ๐2 โ๐3๐0 ๐3 โ๐2โ๐3 ๐0 ๐1๐2 โ๐1 ๐0
๐ค1
๐ค2
๐ค3
The relationship between the euler angles and the euler
parameters is
๐0 = cos๐ + ๐
2cos
๐
2, ๐1 = cos
๐ โ ๐
2sin
๐
2
๐2 = sin๐ โ ๐
2sin
๐
2, ๐3 = sin
๐ + ๐
2cos
๐
2
Numerical Simulations
Question: How to propagate the rigid body?
Answer: There are three ways. But not all of them work
good in numerical simulation.
1) Linear velocity
2) Euler Angles
3) Euler Parameters
Numerical Simulations
Propagate by linear velocity๐๐
๐๐ก= ๐ฃ = ๐ค ร ๐
๐๐ โ ๐๐+1
Directly use the last step location information.
Propagate by euler angles or euler parameter
1) Implemented by rotation matrix.
2) The order of propagate matters.
3) e.g. ฮ๐ด โ ๐ด = ฮ๐ตฮ๐ถฮ๐ท โ ๐ต๐ถ๐ท โ ฮ๐ต โ ๐ต ฮ๐ถ โ ๐ถ ฮ๐ท โ ๐ท
Inverse back to the initial location by the old parameters
and propagate by the new parameters.
Numerical Simulations
Update angular velocity by the Eulerโs equation of
motion with 4-th order Runge-Kutta method.
Initial angular velocity: ๐ค1 = 1,๐ค2 = 0,๐ค3 = 1
Principle Moment of Inertia:
๐ผ1 = ๐ผ2 = 0.6375, ๐ผ3 = 0.075
Analytic Solution
๐ค1 = cos(๐ผ3 โ ๐ผ1๐ผ1
๐ค3๐ก)
๐ค2 = sin(๐ผ3 โ ๐ผ1๐ผ1
๐ค3๐ก)
Numerical Simulations
Numerical Simulationspropagate by linear velocity
If the rigid body is propagated in the tangent direction,
numerical simulations showed the fact that the rigid
body will become larger and larger.
Solution: propagate in the secant direction.
Numerical Simulationspropagate by linear velocity
Numerical Simulationspropagate by linear velocity
Numerical Simulationspropagate by euler angles
Use the relationship between the euler angles and the
angular velocity to update the euler angles in each time
step.
This keeps the shape of the rigid body. However, the
matrix has a big probability to be singular when ๐ = ๐๐.
Example: if we consider the initial position is where the
three euler angles are all zero, then ๐ = 0 and this leads
to a singular matrix.
Solution: no solution.
Numerical Simulationspropagate by euler parameters
Use the relationship between the euler parameters and the angular velocity to update the euler parameters in each step.
This keeps the shape of the rigid body. Also, it is shown in numerical simulation that the matrix can never be singular.
Based on the initial euler angles ๐ = ๐ = ๐ = 0 and the relationship between euler parameters and euler angles, the initial values of the four euler parameters are
๐0 = 1, ๐1 = ๐2 = ๐3 = 0
Numerical Simulationspropagate by euler parameters
Numerical Simulationspropagate by euler parameters
Numerical Simulationspropagate by euler parameters
Numerical Simulationspropagate by euler parameters
Future Work
different shapes of rigid body
outside force to the rigid body
fluid field in the simulation
parachutists motion in the parachute inflation
References
Classical Mechanics, Herbert Goldstein, Charles Poole anb John Safko
http://www.mathworks.com/help/aeroblks/6dofeuleran
gles.html
http://mathworld.wolfram.com/EulerParameters.html
http://www.u.arizona.edu/~pen/ame553/Notes/Lesson%
2010.pdf
Wikipedia