Transcript PPT - Modeling & Simulation Lab.

Physics Based Modeling
Lecture 1
Kwang Hee Ko
Gwangju Institute of Science and
Technology
Introduction
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What is “Physics-based Modeling”???
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The behavior and form of many objects are determined
by the objects’ gross physical property.
This modeling technique uses “physics properties” to
determine the shape and motions of objects.
Constraint-Based Modeling
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Constraints are those which the parts of a model are
supposed to satisfy.
Example 1: A model for human skeletons
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Constraints on connectivity of bones, limits of angular
motion on joints, etc.
Example 2: A sphere on a table
Introduction
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Constraint-Based Modeling
Motivations
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Natural phenomena are characterized by physical
laws.
Deformation is ubiquitous ranging from microscale to nano-scale.
Graphics/animation aims to model and simulate
physical worlds.
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Every component of graphics is relevant to physical
laws.
Physics-based modeling gives rise to a large
variety of applications in graphics, geometric
design, visualization, simulation, etc.
Physics-based modeling has been a very
powerful tool to tackle many real problems.
Applications of Physics-based
Modeling
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Geometric modeling using physics and
energy
Interactive and dynamic editing
Geometric processing
Virtual surgery simulation
Haptic interface
Realistic rendering of natural phenomena
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Fire, wave, etc.
Applications of Physics-based
Modeling
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Flow Simulation
Applications of Physics-based
Modeling
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Various Natural Phenomena
Applications of Physics-based
Modeling
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Fluid Simulation
Applications of Physics-based
Modeling
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Motion Animation/Synthesis
Applications of Physics-based
Modeling
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Virtual Surgery
Applications of Physics-based
Modeling
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Deformation Simulation
What we are going to study…
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We will be studying concepts on
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Physics on rigid and deformable bodies.
Physics on animation
A Bit of Mathematics….
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Quaternion
Calculus
…
Quaternion
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Basics of Quaternion
Application to Rotation
Geometric Transformations
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Rotation is defined by an axis
and an angle of rotation.
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Rotation in 3D is not as
simple as translation.
It can be defined in many
ways.
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Quaternions
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The second rotational modality is rotation
defined by Euler’s theorem and implemented
with quaternions.
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Euler’s rotational theorem
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An arbitrary rotation may be described by only three
parameters.
Historical Backgrounds
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Quaternions were invented by Sir William Rowan Hamilton in 1843.
His aim was to generalize complex numbers to three dimensions.
 Numbers of the form a+ib+jc, where a,b,c are real numbers and
i2=j2=-1.
 He never succeeded in making this generalization.
It has later been proven that the set of three-dimensional numbers is
not closed under multiplication.
Four numbers are needed to describe a rotation followed by a scaling.
 One number describes the size of the scaling.
 One number describes the number of degrees to be rotated.
 Two numbers give the plane in which the vector should be rotated.
Basic Quaternion Mathematics
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Quaternions, denoted q, consist of a scalar part s
and a vector part v=(x,y,z). We will use the
following form.
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Let i2=j2=k2=ijk=-1, ij=k and ji=-k.
A quaternion q can be written:
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q = [s,v] = [s,(x,y,z)] = s+ix+jy+kz.
The addition operator, +, is defined
Basic Quaternion Mathematics
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Multiplication is defined:
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Multiplication by a scalar is defined by
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rq ≡ [r,0]q
Subtraction is defined
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Quaternion multiplication is not generally commutative.
q – q’ ≡ q + (-1)q’
Let q be a quaternion. Then q* is called the conjugate of
q and is defined by q* ≡ [s,v]* ≡ [s, -v].
Basic Quaternion Mathematics
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Let p,q be quaternions. Then
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The norm of a quaternion q.
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||q|| = √qq*
The inner product is defined
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(q*)* = q, (pq)* = q*p*, (p+q)* = p* + q*, qq* = q*q
q·q’ = ss’+v·v’ = ss’ + xx’ + yy’ + zz’
Let q,q’ be quaternions. Define them as the
corresponding four-dimensional vectors and let α be the
angle between them.
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q·q’ = ||q|| ||q’|| cos α .
Basic Quaternion Mathematics
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The unique neutral element under quaternion
multiplication
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I = [1,0]
Inverse under quaternion multiplication
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qq-1=q-1q=I.
q-1=q*/||q||2
Basic Quaternion Mathematics
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Unit quaternions
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If ||q|| = 1, then q is called a unit quaternion.
Use H1 to denote the set of unit quaternions
Let q = [s,v], a unit quaternion. Then, there exists v’
and θ such that q = [cosθ ,v’sinθ ].
Let q, q’ be unit quaternions. Then
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||qq’|| = 1
q-1 = q*
Etc…
Rotation with Quaternions
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Let q=[cosθ,nsinθ] be a unit quaternion. Let r =
(x,y,z) and p[0,r] be a quaternion. Then
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p’= qpq-1 is p rotated 2θ about the axis n.
• Any general three-dimensional rotation
about n, |n|=1 can be obtained by a unit
quaternion.
• Choose q such that q=[cosθ/2,nsinθ/2]
Rotation with Quaternions
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Let q1, q2 be unit quaternions. Rotation by q1
followed by rotation by q2 is equivalent to
rotation by q2q1.
Geometric intuition
Comparison of Quaternions, Euler
Angles and Matrices
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Euler Angles/Matrices – Disadvantages
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Lack of intuition
The order of rotation axes is important.
Gimbal lock
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It is a concept originating from the air and space industry,
where gyroscopes are used.
At a certain situation, two rotations act about the same axis.
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Mathematically gimbal lock corresponds to loosing a degree of
freedom in the general rotation matrix.
Comparison of Quaternions, Euler
Angles and Matrices
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Euler Angles/Matrices – Disadvantages
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Gimbal lock
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If we letβ=π/2, then a rotation with αwill have the same effect as
applying the same rotation with -γ.
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The rotation only depends on the difference and therefore it has
only one degree of freedom. For β=π/2 changes of α and γ result in
rotations about the same axis.
Comparison of Quaternions, Euler
Angles and Matrices
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Euler Angles/Matrices – Disadvantages
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Implementing interpolation is difficult
Ambiguous correspondence to rotations
The result of composition is not apparent
The representation is redundant
Euler Angles/Matrices – Advantages
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The mathematics is well-known and that matrix
applications are relatively easy to implement.
Comparison of Quaternions, Euler
Angles and Matrices
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Quaternions – Disadvantages
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Quaternions only represent rotation
Quaternion mathematics appears complicated
Quaternions – Advantages
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Obvious geometrical interpretation
Coordinate system independency
Simple interpolation methods
Compact representation
No gimbal lock
Simple composition
Interpolation of Solid Orientations
Rigid Body (Single Particle)
Newtonian Mechanics (Single Particle)
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Newton’s Laws
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A body remains at rest or in uniform motion
unless acted upon by a force.
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A body acted upon by a force moves in such a
manner that the time rate of change of
momentum equals the force.
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Force equilibrium
P = mv. F = dP / dt = d(mv)/dt
If two bodies exert forces on each other, these
forces are equal in magnitude and opposite in
direction.
Rigid Body (Single Particle)
Newtonian Mechanics (Single Particle)
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Inertial Frame
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The laws of motion to have meaning, the
motion of bodies must be measured relative to
some reference frame.
Equation of Motion for a Particle.
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F = d(mv)/dt = m dv/dt = mr’’
Example
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Projectile motion in two dimensions.
Rigid Body (Single Particle)
Newtonian Mechanics (Single Particle)
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Question
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No air resistance.
Muzzle velocity of the projectile: v0.
Angle of elevation Θ.
Calculate the projectile’s displacement, velocity
and range.
Rigid Body (Single Particle)
Newtonian Mechanics (Single Particle)
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Solution.
Rigid Body (Single Particle)
Newtonian Mechanics (Single Particle)
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Conservation Theorems
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The total linear momentum P of a particle is
conserved when the total force on it is zero.
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The angular momentum of a particle subject to
no torque is conserved.
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P·s = constant
L = r X P.
The total energy E of a particle in a
conservative force field is a constant in time.
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E = T(kinetic) + U(potential)
Rigid Body (Single Particle)
Newtonian Mechanics (Single Particle)
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Example 2: A cylinder on a curved surface
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A cylinder of mass m and radius R1 rolling
without slippage on a curved surface of radius
R.
Rigid Body (Single Particle)
Newtonian Mechanics (Single Particle)
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Solution
dE/dt = dT/dt + dU/dt = 0
Geometry of Deformable Models
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The models are 3D solids in space.
Global Deformations
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The reference shape s is defined as
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T: global deformation
e: geometric primitive defined parametrically in
u and parametrized by the variables ai.
The vector of global deformation
parameters
Local Deformations
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The displacement d anywhere within a
deformable model is represented as a
linear combination of an infinite number of
basis functions bj(u)
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The diagonal matrix Si is formed form the basis
functions and where qd,I are local degrees of
freedom.
Kinematics and Dynamics
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Kinematic and dynamic formulation of the
deformable models
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Kinematic formulation: the computation of a
Jacobian matrix L
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It allows the transformation of 3D vectors into qdimensional vectors.
Dynamic formulation: based on Lagrangian
dynamics and generalized coordinates.
Kinematics
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The velocity of a point on the model
Kinematics
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Computation of R and B using Quaternions
The dual matrix of the position
vector p(u) = (p1,p2,p3)T
Dynamics
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Lagrange Equations of Motion
Dynamics
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Kinetic Energy: Mass Matrix
Dynamics
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Acceleration and Inertial Forces
Dynamics
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Acceleration and Inertial Forces
Dynamics
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Damping Matrix: Energy Dissipation
Dynamics
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Stiffness Matrix: Strain Energy
Dynamics
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Stiffness Matrix: Strain Energy
Dynamics
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External Forces