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Rigid Body Motion?  Answer: in Total, There Are 6 Degrees of Freedom

Rigid Body Motion?  Answer: in Total, There Are 6 Degrees of Freedom

Rigid Body

Speaker: Xiaolei Chen Advisor: Prof. Xiaolin Li Department of Applied and Statistics Stony Brook University (SUNY) Content

 Introduction  Angular  Angular and Tensor  Euler’s Equations of and Euler Parameters  Numerical Simulation  Future Introduction

 Question: How many degrees of freedom in the 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 velocity  Rotational Motion: , euler angles, euler parameters Angular Velocity

In , the angular velocity is defined as the rate of change of angular . In 2D cases, the angular velocity is a scalar.

푑휃 Equation: = 푤 푑푡

Linear velocity: 푣 = 푤 × 푟 Angular Velocity

In the 3D rigid body rotation, the angular velocity is a vector, and use 푤 = (푤1, 푤2, 푤3) to denote it.

 In physics, the angular momentum (퐿) is a measure of the amount of rotation an object has, taking into account its mass, shape and .  The 퐿 of a particle about a given origin in the body axes is given by 퐿 = 푟 × 푝 = 푟 × (푚푣) where 푝 is the linear momentum, and 푣 is the linear velocity.  Linear velocity: 푣 = 푤 × 푟 퐿 = 푚푟 × 푤 × 푟 = 푚(푤푟2 − 푟(푟 ∙ 푤)) Angular Momentum

 Each component of the angular Momentum is 2 2 퐿1 = 푚(푟 − 푥 )푤1 − 푚푥푦푤2 − 푚푥푧푤3 2 2 퐿2 = −푚푦푥푤1 + 푚(푟 − 푦 )푤2 − 푚푦푧푤3 2 2 퐿3 = −푚푧푥푤1 − 푚푧푦푤2 + 푚(푟 − 푧 )푤3  It can be written in the tensor formula. 퐿 = 퐼푤 퐼푥푥 퐼푥푦 퐼푥푧 2 퐼 = 퐼푦푥 퐼푦푦 퐼푦푧 , 퐼푎푏 = 푚(푟 훿푎푏 − 푎푏) 퐼푧푥 퐼푧푦 퐼푧푧 Inertia Tensor

of inertia is the mass property of a rigid body that determines the needed for a desired angular about an axis of rotation. It can be defined by the Inertia Tensor (퐼), which consists of nine components (3 × 3 matrix).  In continuous cases,

2 퐼푎푏 = 휌 푉 푟 훿푎푏 − 푎푏 푑푉 푉  The inertia tensor is constant in the body axes. Inertia Tensor Euler’s

 Apply the eigen-decomposition to the inertial tensor and obtain 푇 퐼 = 푄Λ푄 , Λ = 푑푖푎푔 퐼1, 퐼2, 퐼3 where 퐼푖 are called principle moments of inertia, and the columns of 푄 are called principle axes. The rigid body rotates in angular velocity (푤푥, 푤푦, 푤푧) with respect to the principle axes.  Specially, consider the cases where 퐼 is diagonal. 퐼 = 푑푖푎푔 퐼1, 퐼2, 퐼3 , 퐿푖 = 퐼푖푤푖, 푖 = 1,2,3 The principle axes are the same as the 푥, 푦, 푧 axes in the body axes. Euler’s Equations of Motion

 The rate of change of a vector 퐺 in the axes can be written as 푑퐺 푑퐺 = + 푤 × 퐺 푑푡 푑푡 푠 푏 푑퐺 where is the time rate of change of the vector 퐺 in 푑푡 푏 the body axes.  Apply the formula to the angular momentum 퐿 푑퐿 푑퐿 = + 푤 × 퐺 푑푡 푑푡 푠 푏 Euler’s Equations of Motion

 Use the relationship 퐿 = 퐼푤 to obtain 푑(퐼푤) 푑퐿 + 푤 × 퐿 = = 휏 푑푡 푑푡 푠 The second equality is based on the definition of torque. 푑퐿 푑(푟 × 푝) 푑푝 = = 푟 × = 푟 × 퐹 = 휏 푑푡 푑푡 푑푡  Euler’s equation of motion

퐼1푤 1 − 푤2푤3 퐼2 − 퐼3 = 휏1 퐼2푤 2 − 푤3푤1 퐼3 − 퐼1 = 휏2 퐼3푤 3 − 푤1푤2 퐼1 − 퐼2 = 휏3 Euler Angles

 The Euler angles are three angles introduced to describe the of a rigid body.  Usually, use ϕ, 휃, 휓 to denote them. Euler Angles

The order of the rotation steps matters. Euler Angles

 Three rotation matrices are 푐표푠휙 푠푖푛휙 0 퐷 = −푠푖푛휙 푐표푠휙 0 0 0 1

1 0 0 퐶 = 0 푐표푠휃 푠푖푛휃 0 −푠푖푛휃 푐표푠휃

푐표푠휓 푠푖푛휓 0 퐵 = −푠푖푛휓 푐표푠휓 0 0 0 1 Euler Angles

Then, the is 푐표푠휓푐표푠휙 − 푐표푠휃푠푖푛휙푠푖푛휓 푐표푠휓푠푖푛휙 + 푐표푠휃푐표푠휙푠푖푛휓 푠푖푛휓푠푖푛휃 퐴 = 퐵퐶퐷 = −푠푖푛휓푐표푠휙 − 푐표푠휃푠푖푛휙푐표푠휓 −푠푖푛휓푠푖푛휙 + 푐표푠휃푐표푠휙푐표푠휓 푐표푠휓푠푖푛휃 푠푖푛휃푠푖푛휙 −푠푖푛휃푐표푠휙 푐표푠휃 This matrix can be used to convert the coordinates in the space axes to the coordinates in the body axes. The inverse is 퐴−1 = 퐴푇. Assume 푥 is the coordinates in the space axes, and 푥′ is the coordinates in the body axes. 푥′ = 퐴푥 푥 = 퐴푇푥′ Euler Angles

The relationship between the angular velocity and the euler angles is

푤1 0 휃 0 휙 푇 푇 푇 −1 푤2 = 0 + 퐷 0 + 퐷 퐶 0 = 퐽 휃 푤3 휙 0 휓 휓 0 푐표푠휙 푠푖푛휃푠푖푛휙 퐽−1 = 0 푠푖푛휙 −푠푖푛휃푐표푠휙 1 0 푐표푠휃 휙 −푠푖푛휙푐표푡휃 푐표푠휙푐표푡휃 1 푤1 휃 = 푐표푠휙 푠푖푛휙 0 푤2 휓 푠푖푛휙푐푠푐휃 −푐표푠휙푐푠푐휃 0 푤3 Euler Parameters

The four parameters 푒0, 푒1, 푒2, 푒3 describe a finite about an arbitrary axis. The parameters are defined as

휑 Def: 푒 = 푐표푠 0 2 휑 And 푒 , 푒 , 푒 = 푛sin( ) 1 2 3 2 Euler Parameters

In this representation, 푛 is the unit normal vector, and 휑 is the rotation angle. 2 2 2 2 Constraint: 푒0 + 푒1 + 푒2 + 푒3 = 1

2 2 2 2 푒0 + 푒1 − 푒2 − 푒3 2(푒1푒2 + 푒0푒3) 2(푒1푒3 − 푒0푒2) 2 2 2 2 퐴 = 2(푒1푒2 − 푒0푒3) 푒0 − 푒1 + 푒2 − 푒3 2(푒2푒3 + 푒0푒1) 2 2 2 2 2(푒1푒3 + 푒0푒2) 2(푒2푒3 − 푒0푒1) 푒0 − 푒1 + 푒2 − 푒3 Euler Parameters

The relationship between the angular velocity and the euler parameters is 푒 0 −푒1 −푒2 −푒3 푤 1 1 푒1 푒0 푒3 −푒2 푤 = −푒 푒 푒 2 푒 2 2 3 0 1 푤 푒 −푒 푒 3 푒 3 2 1 0 The relationship between the euler angles and the euler parameters is 휙 + 휓 휃 휙 − 휓 휃 푒 = cos cos , 푒 = cos sin 0 2 2 1 2 2 휙 − 휓 휃 휙 + 휓 휃 푒 = sin sin , 푒 = sin cos 2 2 2 3 2 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 velocity 푑푟 = 푣 = 푤 × 푟 푑푡 푟푛 → 푟푛+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. Δ퐴 ∙ 퐴 = Δ퐵Δ퐶Δ퐷 ∙ 퐵퐶퐷 ≠ Δ퐵 ∙ 퐵 Δ퐶 ∙ 퐶 Δ퐷 ∙ 퐷 Inverse back to the initial location by the old parameters and propagate by the new parameters. Numerical Simulations

 Update angular velocity by the Euler’s equation of motion with 4-th order Runge-Kutta method.

 Initial angular velocity: 푤1 = 1, 푤2 = 0, 푤3 = 1  Principle : 퐼1 = 퐼2 = 0.6375, 퐼3 = 0.075  Analytic Solution 퐼3 − 퐼1 푤1 = cos( 푤3푡) 퐼1 퐼3 − 퐼1 푤2 = sin( 푤3푡) 퐼1 Numerical Simulations Numerical Simulations propagate 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 Simulations propagate by linear velocity Numerical Simulations propagate by linear velocity Numerical Simulations propagate 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 휃 = 푛휋.  Example: if we consider the initial is where the three euler angles are all zero, then 휃 = 0 and this leads to a singular matrix.  Solution: no solution. Numerical Simulations propagate 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 휙 = 휃 = 휑 = 0 and the relationship between euler parameters and euler angles, the initial values of the four euler parameters are 푒0 = 1, 푒1 = 푒2 = 푒3 = 0 Numerical Simulations propagate by euler parameters Numerical Simulations propagate by euler parameters Numerical Simulations propagate by euler parameters Numerical Simulations propagate by euler parameters Future Work

 different shapes of rigid body  outside to the rigid body  fluid in the simulation  parachutists motion in the parachute inflation References

 Classical , 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