Rigid Body Motionchenx/notes/RGB.pdfΒ Β· columns of πare called principle axes. The rigid body...
Transcript of Rigid Body Motionchenx/notes/RGB.pdfΒ Β· columns of πare called principle axes. The rigid body...
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Rigid Body Rotation
Speaker: Xiaolei Chen Advisor: Prof. Xiaolin Li
Department of Applied Mathematics and Statistics
Stony Brook University (SUNY)
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Content
Introduction
Angular Velocity
Angular Momentum and Inertia Tensor
Eulerβs Equations of Motion
Euler Angles and Euler Parameters
Numerical Simulation
Future work
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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
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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: ππ
ππ‘= π€
Linear velocity: π£ = π€ Γ π
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Angular Velocity
In the 3D rigid body rotation, the angular velocity is a
vector, and use π€ = (π€1, π€2, π€3) to denote it.
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Angular Momentum
In physics, the angular momentum (πΏ) is a measure of the
amount of rotation an object has, taking into account its
mass, shape and speed.
The πΏ of a particle about a given origin in the body axes is
given by
πΏ = π Γ π = π Γ (ππ£)
where π is the linear momentum, and π£ is the linear velocity.
Linear velocity: π£ = π€ Γ ππΏ = ππ Γ π€ Γ π = π(π€π2 β π(π β π€))
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Angular Momentum
Each component of the angular Momentum is
πΏ1 = π(π2 β π₯2)π€1 βππ₯π¦π€2 βππ₯π§π€3
πΏ2 = βππ¦π₯π€1 +π(π2 β π¦2)π€2 βππ¦π§π€3
πΏ3 = βππ§π₯π€1 βππ§π¦π€2 +π(π2 β π§2)π€3
It can be written in the tensor formula.πΏ = πΌπ€
πΌ =
πΌπ₯π₯ πΌπ₯π¦ πΌπ₯π§πΌπ¦π₯ πΌπ¦π¦ πΌπ¦π§πΌπ§π₯ πΌπ§π¦ πΌπ§π§
, πΌππ = π(π2πΏππ β ππ)
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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 (πΌ), which consists of nine
components (3 Γ 3 matrix).
In continuous cases,
πΌππ = π
π π π2πΏππ β ππ ππ
The inertia tensor is constant in the body axes.
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Inertia Tensor
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Eulerβs Equations of Motion
Apply the eigen-decomposition to the inertial tensor and
obtain
πΌ = πΞππ , Ξ = ππππ πΌ1, πΌ2, πΌ3
where πΌπ are called principle moments of inertia, and the
columns of π are called principle axes. The rigid body rotates in angular velocity (π€π₯, π€π¦, π€π§) with respect to the
principle axes.
Specially, consider the cases where πΌ is diagonal.
πΌ = ππππ πΌ1, πΌ2, πΌ3 , πΏπ = πΌππ€π , π = 1,2,3
The principle axes are the same as the π₯, π¦, π§ axes in the
body axes.
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Eulerβs Equations of Motion
The time rate of change of a vector πΊ in the space axes
can be written asππΊ
ππ‘π
=ππΊ
ππ‘π
+π€ Γ πΊ
where ππΊ
ππ‘ πis the time rate of change of the vector πΊ in
the body axes.
Apply the formula to the angular momentum πΏππΏ
ππ‘π
=ππΏ
ππ‘π
+π€ Γ πΊ
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Eulerβs Equations of Motion
Use the relationship πΏ = πΌπ€ to obtainπ(πΌπ€)
ππ‘+ π€ Γ πΏ =
ππΏ
ππ‘π
= π
The second equality is based on the definition of torque.ππΏ
ππ‘=π(π Γ π)
ππ‘= π Γ
ππ
ππ‘= π Γ πΉ = π
Eulerβs equation of motion
πΌ1 π€1 βπ€2π€3 πΌ2 β πΌ3 = π1πΌ2 π€2 βπ€3π€1 πΌ3 β πΌ1 = π2πΌ3 π€3 βπ€1π€2 πΌ1 β πΌ2 = π3
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Euler Angles
The Euler angles are three angles introduced to describe
the orientation of a rigid body.
Usually, use Ο, π, π to denote them.
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The order of the rotation steps matters.
Euler Angles
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Euler Angles
Three rotation matrices are
π· =πππ π π πππ 0βπ πππ πππ π 0
0 0 1
πΆ =1 0 00 πππ π π πππ0 βπ πππ πππ π
π΅ =πππ π π πππ 0βπ πππ πππ π 0
0 0 1
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Then, the rotation matrix is
π΄ = π΅πΆπ· =
πππ ππππ π β πππ ππ ππππ πππ πππ ππ πππ + πππ ππππ ππ πππ π ππππ πππβπ ππππππ π β πππ ππ ππππππ π βπ ππππ πππ + πππ ππππ ππππ π πππ ππ πππ
π ππππ πππ βπ ππππππ π πππ π
This matrix can be used to convert the coordinates in the
space axes to the coordinates in the body axes.
The inverse is π΄β1 = π΄π.
Assume π₯ is the coordinates in the space axes, and π₯β² is the
coordinates in the body axes.
π₯β² = π΄π₯π₯ = π΄ππ₯β²
Euler Angles
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Euler Angles
The relationship between the angular velocity and the
euler angles isπ€1
π€2
π€3
=00 π+ π·π
π00
+ π·ππΆπ00 π= π½β1
π π π
π½β1 =0 πππ π π ππππ πππ0 π πππ βπ ππππππ π1 0 πππ π
π π π
=
βπ ππππππ‘π πππ ππππ‘π 1πππ π π πππ 0
π πππππ ππ βπππ πππ ππ 0
π€1
π€2
π€3
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Euler Parameters
The four parameters π0, π1, π2, π3 describe a finite about an arbitrary axis. The parameters are defined as
Def: π0 = πππ π
2
And π1, π2, π3 = πsin(π
2)
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Euler Parameters
In this representation, π is the unit normal vector, and π is the rotation angle.
Constraint: π02 + π1
2 + π22 + π3
2 = 1
π΄ =
π02 + π1
2 β π22 β π3
2 2(π1π2 + π0π3) 2(π1π3 β π0π2)
2(π1π2 β π0π3) π02 β π1
2 + π22 β π3
2 2(π2π3 + π0π1)
2(π1π3 + π0π2) 2(π2π3 β π0π1) π02 β π1
2 + π22 β π3
2
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Euler Parameters
The relationship between the angular velocity and the
euler parameters is π0 π1 π2 π3
=1
2
βπ1 βπ2 βπ3π0 π3 βπ2βπ3 π0 π1π2 βπ1 π0
π€1
π€2
π€3
The relationship between the euler angles and the euler
parameters is
π0 = cosπ + π
2cos
π
2, π1 = cos
π β π
2sin
π
2
π2 = sinπ β π
2sin
π
2, π3 = sin
π + π
2cos
π
2
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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
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Numerical Simulations
Propagate by linear velocityππ
ππ‘= π£ = π€ Γ π
ππ β ππ+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. Ξπ΄ β π΄ = Ξπ΅ΞπΆΞπ· β π΅πΆπ· β Ξπ΅ β π΅ ΞπΆ β πΆ Ξπ· β π·
Inverse back to the initial location by the old parameters
and propagate by the new parameters.
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Numerical Simulations
Update angular velocity by the Eulerβs equation of
motion with 4-th order Runge-Kutta method.
Initial angular velocity: π€1 = 1,π€2 = 0,π€3 = 1
Principle Moment of Inertia:
πΌ1 = πΌ2 = 0.6375, πΌ3 = 0.075
Analytic Solution
π€1 = cos(πΌ3 β πΌ1πΌ1
π€3π‘)
π€2 = sin(πΌ3 β πΌ1πΌ1
π€3π‘)
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Numerical Simulations
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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.
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Numerical Simulationspropagate by linear velocity
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Numerical Simulationspropagate by linear velocity
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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 π = ππ.
Example: if we consider the initial position is where the
three euler angles are all zero, then π = 0 and this leads
to a singular matrix.
Solution: no solution.
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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 π = π = π = 0 and the relationship between euler parameters and euler angles, the initial values of the four euler parameters are
π0 = 1, π1 = π2 = π3 = 0
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Numerical Simulationspropagate by euler parameters
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Numerical Simulationspropagate by euler parameters
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Numerical Simulationspropagate by euler parameters
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Numerical Simulationspropagate by euler parameters
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Future Work
different shapes of rigid body
outside force to the rigid body
fluid field in the simulation
parachutists motion in the parachute inflation
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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