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 Euler’s 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. 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. Angular Momentum In physics, the angular momentum (퐿) is a measure of the amount of rotation an object has, taking into account its mass, shape and speed. 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 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 (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 Equations of Motion 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 time rate of change of a vector 퐺 in the space 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 orientation 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 rotation matrix 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 Moment of Inertia: 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 position 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 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.