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

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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.
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    Rotation: Moment of Inertia and Torque Every time we push a door open or tighten a bolt using a wrench, we apply a force that results in a rotational motion about a fixed axis. Through experience we learn that where the force is applied and how the force is applied is just as important as how much force is applied when we want to make something rotate. This tutorial discusses the dynamics of an object rotating about a fixed axis and introduces the concepts of torque and moment of inertia. These concepts allows us to get a better understanding of why pushing a door towards its hinges is not very a very effective way to make it open, why using a longer wrench makes it easier to loosen a tight bolt, etc. This module begins by looking at the kinetic energy of rotation and by defining a quantity known as the moment of inertia which is the rotational analog of mass. Then it proceeds to discuss the quantity called torque which is the rotational analog of force and is the physical quantity that is required to changed an object's state of rotational motion. Moment of Inertia Kinetic Energy of Rotation Consider a rigid object rotating about a fixed axis at a certain angular velocity. Since every particle in the object is moving, every particle has kinetic energy. To find the total kinetic energy related to the rotation of the body, the sum of the kinetic energy of every particle due to the rotational motion is taken. The total kinetic energy can be expressed as ..
  • Chapter 1 Rigid Body Kinematics

    Chapter 1 Rigid Body Kinematics

    Chapter 1 Rigid Body Kinematics 1.1 Introduction This chapter builds up the basic language and tools to describe the motion of a rigid body – this is called rigid body kinematics. This material will be the foundation for describing general mech- anisms consisting of interconnected rigid bodies in various topologies that are the focus of this book. Only the geometric property of the body and its evolution in time will be studied; the re- sponse of the body to externally applied forces is covered in Chapter ?? under the topic of rigid body dynamics. Rigid body kinematics is not only useful in the analysis of robotic mechanisms, but also has many other applications, such as in the analysis of planetary motion, ground, air, and water vehicles, limbs and bodies of humans and animals, and computer graphics and virtual reality. Indeed, in 1765, motivated by the precession of the equinoxes, Leonhard Euler decomposed a rigid body motion to a translation and rotation, parameterized rotation by three principal-axis rotation (Euler angles), and proved the famous Euler’s rotation theorem [?]. A body is rigid if the distance between any two points fixed with respect to the body remains constant. If the body is free to move and rotate in space, it has 6 degree-of-freedom (DOF), 3 trans- lational and 3 rotational, if we consider both translation and rotation. If the body is constrained in a plane, then it is 3-DOF, 2 translational and 1 rotational. We will adopt a coordinate-free ap- proach as in [?, ?] and use geometric, rather than purely algebraic, arguments as much as possible.