Rigid Body Dynamics CSE169: Computer Animation Instructor: Steve - PowerPoint PPT Presentation

Rigid Body Dynamics CSE169: Computer Animation Instructor: Steve Rotenberg UCSD, Spring 2016

Cross Product i j k   a b a a a x y z b b b x y z        a b a b a b a b a b a b a b y z z y z x x z x y y x

Properties of the Cross Product  Non-commutative: 𝐛 × 𝐜 ≠ 𝐜 × 𝐛  Non-associative: 𝐛 × 𝐜 × 𝐝 ≠ 𝐛 × 𝐜 × 𝐝

Cross Product        a b a b a b a b a b a b a b y z z y z x x z x y y x   c a b     c 0 b a b a b x x z y y z     c a b 0 b a b y z x y x z      c a b a b 0 b z z x x y z

Cross Product     c 0 b a b a b x x z y y z     c a b 0 b a b y z x y x z      c a b a b 0 b z z x x y z        c 0 a a b x z y x          c a 0 a b       y z x y            c a a 0 b   z y x z

Cross Product        c 0 a a b x z y x          c a 0 a b       y z x y            c a a 0 b   z y x z    ˆ a b a b    0 a a z y     ˆ a a 0 a   z x    a a 0   y x

Hat Operator  We’ve introduced the ‘hat’ operator which converts a 𝑈 = −𝐛 vector into a skew-symmetric matrix 𝐛  This allows us to turn a cross product of two vectors into a dot product of a matrix and a vector  This is mainly for algebraic convenience, as the dot product is associative (although still not commutative) ∙ 𝐜 = 𝐛 × 𝐜 𝐛 ∙ 𝐜 ≠ 𝐜 ∙ 𝐛 𝐛 ∙ 𝐝 = 𝐛 𝐛 ∙ 𝐜 ∙ 𝐜 ∙ 𝐝

Derivative of a Rotating Vector  Let’s say that vector r is rotating around the origin, maintaining a fixed distance  At any instant, it has an angular velocity of ω ω r d r   ω r dt ω  r

Derivative of Rotating Matrix  If matrix A is a rigid 3x3 matrix rotating with angular velocity ω  This implies that the a , b , and c axes must be rotating around ω  The derivatives of each axis are ω x a , ω x b , and ω x c , and so the derivative of the entire matrix is: d A     ˆ ω ω A A dt

Product Rule  The product rule defines the derivative of products   d ab da db   b a dt dt dt   d abc da db dc    bc a c ab dt dt dt dt

Product Rule  It can be extended to vector and matrix products as well    d a b d a d b     b a dt dt dt    a b a b d d d     b a dt dt dt    d A B d A d B     B A dt dt dt

Eigenvalue Equation Lets say we have a known matrix M and we want to know if there is  any vector x and scalar s such that 𝐍𝐲 = 𝑡𝐲 This is known as an eigenvalue equation, and for a NxN matrix,  there should be up to N eigenvectors 𝐲 𝑗 and N eigenvalues 𝑡 𝑗 that satisfy the equation If M is a symmetric matrix (i.e., 𝐍 𝑈 = 𝐍 ) then all of the eigenvalues  will be real numbers and the eigenvectors will be real, orthonormal vectors (otherwise, some of the eigenvalues/eigenvectors will be complex)

Symmetric Matrix If we have a symmetric matrix 𝐍 , we can diagonalize it:  𝐍 0 = 𝐁 𝑈 ∙ 𝐍 ∙ 𝐁 Where 𝐍 0 is a diagonal matrix and 𝐁 is an orthonormal (pure  rotation) matrix The columns of 𝐁 are the eigenvectors of 𝐍 and the diagonal  elements in 𝐍 0 are the corresponding eigenvalues The symmetric Jacobi algorithm is a simple and effective matrix  algorithm for computing this diagonalization

Symmetric Matrix Diagonalization 𝑁 𝑦𝑦 𝑁 𝑦𝑧 𝑁 𝑦𝑨 𝑁 𝑦𝑧 𝑁 𝑧𝑧 𝑁 𝑧𝑨 𝐍 = 𝑁 𝑦𝑨 𝑁 𝑧𝑨 𝑁 𝑨𝑨 𝑁 𝑦 0 0 𝐍 0 = 𝐁 𝑈 ∙ 𝐍 ∙ 𝐁 𝑥ℎ𝑓𝑠𝑓 𝐍 0 = 0 𝑁 𝑧 0 0 0 𝑁 𝑨

Dynamics of Particles

Kinematics of a Particle x position d x  v veloc ity dt 2 d v d x   a accelerati on 2 dt dt

Mass, Momentum, and Force m mass  p m v momentum d p   f m a force dt

Moment of Momentum  The moment of momentum is a vector   L r p  Also known as angular momentum (the two terms mean basically the same thing, but are used in slightly different situations)  Angular momentum has parallel properties with linear momentum  In particular, like the linear momentum, angular momentum is conserved in a mechanical system  It is typically represented with a capital L , which is unfortunately inconsistent with our standard of using lowercase for vectors…

Moment of Momentum  L is the same for all three of these particles p • p • p r r • 3 2   r L r p 1

Moment of Momentum  L is different for all of these particles p • p • r   1 L r p r 2 r 3 p •

Moment of Force (Torque)  The moment of force (or torque ) about a point is the rate of change of the moment of momentum about that point d L τ  dt

Moment of Force (Torque)   L r p L r p d d d      τ p r dt dt dt     τ v p r f       τ v m v r f   τ r f

Rotational Inertia  L = r x p is a general expression for the moment of momentum of a particle  In a case where we have a particle rotating around the origin while keeping a fixed distance, we can re-express the moment of momentum in terms of it’s angular velocity ω

Rotational Inertia   L r p       L r v r v m m            ω ω L m r r m r r     ˆ ˆ ω L m r r   ω L I    ˆ ˆ I r r m

Rotational Inertia    ˆ ˆ I m r r       0 r r 0 r r z y z y          I m r 0 r r 0 r     z x z x       r r 0 r r 0     y x y x     2 2 r r r r r r y z x y x z       2 2 I m r r r r r r   x y x z y z     2 2 r r r r r r   x z y z x y

Rotational Inertia        2 2 m r r mr r mr r y z x y x z         2 2 I mr r m r r mr r   x y x z y z        2 2 mr r mr r m r r   x z y z x y   ω L I

Rotational Inertia  The rotational inertia matrix I is a 3x3 matrix that is essentially the rotational equivalent of mass  It relates the angular momentum of a system to its angular velocity by the equation   ω L I  This is similar to how mass relates linear momentum to linear velocity, but rotation adds additional complexity  p m v

Systems of Particles

Systems of Particles n   m m tota l mass of all particles total i  i 1  x m  i i x position of center of mass  cm m i     p p m v to tal momentum cm i i i

Velocity of Center of Mass  x m x d d   i i cm v  cm dt dt m i d x   i m v m i dt   i i v   cm m m i i p  cm v cm m total  p m v cm total cm

Force on a Particle  The change in momentum of the center of mass is equal to the sum of all of the forces on the individual particles  This means that the resulting change in the total momentum is independent of the location of the applied force   p p cm i  d p d p d p      i cm i f i dt dt dt

Systems of Particles  The total moment of momentum around the center of mass is:    L r p cm i i       L x x p cm i cm i

Torque in a System of Particles    L r p cm i i   d r p d L   τ i i cm cm dt dt     d r p  τ i i cm dt      τ r f cm i i

Systems of Particles  We can see that a system of particles behaves a lot like a particle itself  It has a mass, position (center of mass), momentum, velocity, acceleration, and it responds to forces   f f cm i  We can also define it’s angular momentum and relate a change in system angular momentum to a force applied to an individual particle      τ r f cm i i