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. Equation: dθ dt = w Linear velocity: v = w r
Angular Velocity In the 3D rigid body rotation, the angular velocity is a vector, and use w = (w 1, w 2, w 3 ) to denote it.
Angular Momentum In physics, the angular momentum (L) is a measure of the amount of rotation an object has, taking into account its mass, shape and speed. The L of a particle about a given origin in the body axes is given by L = r p = r (mv) where p is the linear momentum, and v is the linear velocity. Linear velocity: v = w r L = mr w r = m(wr 2 r(r w))
Angular Momentum Each component of the angular Momentum is L 1 = m(r 2 x 2 )w 1 mxyw 2 mxzw 3 L 2 = myxw 1 + m(r 2 y 2 )w 2 myzw 3 L 3 = mzxw 1 mzyw 2 + m(r 2 z 2 )w 3, I = m(r 2 δ ab) It can be written in the tensor formula. L = Iw I xx I xy I xz I = I yx I yy I yz I zx I zy I zz ab ab
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 (I), which consists of nine components (3 3 matrix). In continuous cases, I ab = V ρ V r 2 δ ab ab dv 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 I = QΛQ T, Λ = diag I 1, I 2, I 3 where I i are called principle moments of inertia, and the columns of Q are called principle axes. The rigid body rotates in angular velocity (w x, w y, w z ) with respect to the principle axes. Specially, consider the cases where I is diagonal. I = diag I 1, I 2, I 3, L i = I i w i, i = 1,2,3 The principle axes are the same as the x, y, z axes in the body axes.
Euler s Equations of Motion The time rate of change of a vector G in the space axes can be written as dg = dg + w G dt s dt b where dg is the time rate of change of the vector G in dt b the body axes. Apply the formula to the angular momentum L dl = dl + w G dt s dt b
Euler s Equations of Motion Use the relationship L = Iw to obtain d(iw) + w L = dl = τ dt dt s The second equality is based on the definition of torque. dl d(r p) = = r dp = r F = τ dt dt dt Euler s equation of motion I 1 I 2 I 3 w 1 w 2 w 3 I 2 I 3 = τ 1 w 2 w 3 w 1 I 3 I 1 = τ 2 w 3 w 1 w 2 I 1 I 2 = τ 3
Euler Angles The Euler angles are three angles introduced to describe the orientation of a rigid body. Usually, use ϕ, θ, ψ to denote them.
Euler Angles The order of the rotation steps matters.
Euler Angles Three rotation matrices are cosφ sinφ 0 D = sinφ cosφ 0 0 0 1 C = B = 1 0 0 0 cosθ sinθ 0 sinθ cosθ cosψ sinψ 0 sinψ cosψ 0 0 0 1
Euler Angles Then, the rotation matrix is A = BCD = cosψcosφ cosθsinφsinψ cosψsinφ + cosθcosφsinψ sinψsinθ sinψcosφ cosθsinφcosψ sinψsinφ + cosθcosφcosψ cosψsinθ sinθsinφ sinθcosφ cosθ This matrix can be used to convert the coordinates in the space axes to the coordinates in the body axes. The inverse is A 1 = A T. Assume x is the coordinates in the space axes, and x is the coordinates in the body axes. x = Ax x = A T x
Euler Angles The relationship between the angular velocity and the euler angles is w 1 0 w 2 = 0 0 φ + D T 0 + D T C T 0 = J 1 θ w 3 φ 0 ψ ψ 0 cosφ sinθsinφ J 1 = 0 sinφ sinθcosφ φ θ ψ = 1 0 cosθ sinφcotθ cosφcotθ 1 cosφ sinφ 0 sinφcscθ cosφcscθ 0 w 1 w 2 w 3
Euler Parameters The four parameters e 0, e 1, e 2, e 3 describe a finite about an arbitrary axis. The parameters are defined as Def: e 0 = cos φ 2 And e 1, e 2, e 3 = nsin( φ 2 )
Euler Parameters In this representation, n is the unit normal vector, and φ is the rotation angle. Constraint: e 0 2 + e 1 2 + e 2 2 + e 3 2 = 1 A = e 2 0 + e 2 1 e 2 2 2 e 3 2(e 1 e 2 + e 0 e 3 ) 2(e 1 e 3 e 0 e 2 ) 2(e 1 e 2 e 0 e 3 ) e 2 0 e 2 1 + e 2 2 2 e 3 2(e 2 e 3 + e 0 e 1 ) 2(e 1 e 3 + e 0 e 2 ) 2(e 2 e 3 e 0 e 1 ) e 2 0 e 2 1 + e 2 2 2 e 3
Euler Parameters The relationship between the angular velocity and the euler parameters is e 0 e 1 e 2 e 3 w e 1 1 e = 0 e 3 e 1 2 w e 2 2 e 3 e 0 e 2 1 w e 3 e 2 e 1 e 3 0 The relationship between the euler angles and the euler parameters is φ + ψ e 0 = cos cos θ 2 2, e φ ψ 1 = cos sin θ 2 2 φ ψ e 2 = sin sin θ 2 2, e φ + ψ 3 = sin cos θ 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 dr = v = w r dt r n r n+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. ΔA A = ΔBΔCΔD BCD ΔB B ΔC C ΔD D 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: w 1 = 1, w 2 = 0, w 3 = 1 Principle Moment of Inertia: I 1 = I 2 = 0.6375, I 3 = 0.075 Analytic Solution w 1 = cos( I 3 I 1 w I 3 t) 1 w 2 = sin( I 3 I 1 w I 3 t) 1
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 θ = nπ. Example: if we consider the initial position is where the three euler angles are all zero, then θ = 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 φ = θ = φ = 0 and the relationship between euler parameters and euler angles, the initial values of the four euler parameters are e 0 = 1, e 1 = e 2 = e 3 = 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