physics based modeling lecture 1 kwang hee ko gwangju institute of science and technology
TRANSCRIPT
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Physics Based Modeling
Lecture 1Kwang Hee KoGwangju Institute of Science and Technology
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Introduction
What is “Physics-based Modeling”??? 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
Constraints are those which the parts of a model are supposed to satisfy.
Example 1: A model for human skeletons Constraints on connectivity of bones, limits of angular
motion on joints, etc. Example 2: A sphere on a table
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Introduction
Constraint-Based Modeling
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Motivations
Natural phenomena are characterized by physical laws.
Deformation is ubiquitous ranging from micro-scale to nano-scale.
Graphics/animation aims to model and simulate physical worlds. 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.
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Applications of Physics-based Modeling Geometric modeling using physics and
energy Interactive and dynamic editing Geometric processing Virtual surgery simulation Haptic interface Realistic rendering of natural
phenomena Fire, wave, etc.
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Applications of Physics-based Modeling Flow Simulation
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Applications of Physics-based Modeling Various Natural Phenomena
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Applications of Physics-based Modeling Fluid Simulation
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Applications of Physics-based Modeling Motion Animation/Synthesis
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Applications of Physics-based Modeling Virtual Surgery
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Applications of Physics-based Modeling Deformation Simulation
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What we are going to study…
We will be studying concepts on Physics on rigid and deformable bodies. Physics on animation
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A Bit of Mathematics….
Quaternion Calculus …
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Quaternion
Basics of Quaternion Application to Rotation
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Geometric Transformations
Rotation is defined by an axis and an angle of rotation.
Rotation in 3D is not as simple as translation.
It can be defined in many ways.
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Quaternions
The second rotational modality is rotation defined by Euler’s theorem and implemented with quaternions. Euler’s rotational theorem
An arbitrary rotation may be described by only three parameters.
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Historical Backgrounds
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.
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Basic Quaternion Mathematics
Quaternions, denoted q, consist of a scalar part s and a vector part v=(x,y,z). We will use the following form. Let i2=j2=k2=ijk=-1, ij=k and ji=-k. A quaternion q can be written:
q = [s,v] = [s,(x,y,z)] = s+ix+jy+kz.
The addition operator, +, is defined
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Basic Quaternion Mathematics
Multiplication is defined:
Quaternion multiplication is not generally commutative.
Multiplication by a scalar is defined by rq ≡ [r,0]q
Subtraction is defined 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].
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Basic Quaternion Mathematics
Let p,q be quaternions. Then (q*)* = q, (pq)* = q*p*, (p+q)* = p* + q*, qq* = q*q
The norm of a quaternion q. ||q|| = √qq*
The inner product is defined 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. q·q’ = ||q|| ||q’|| cos α .
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Basic Quaternion Mathematics
The unique neutral element under quaternion multiplication I = [1,0]
Inverse under quaternion multiplication qq-1=q-1q=I. q-1=q*/||q||2
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Basic Quaternion Mathematics
Unit quaternions 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
||qq’|| = 1 q-1 = q*
Etc…
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Rotation with Quaternions
Let q=[cosθ,nsinθ] be a unit quaternion. Let r = (x,y,z) and p[0,r] be a quaternion. Then 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]
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Rotation with Quaternions
Let q1, q2 be unit quaternions. Rotation by q1 followed by rotation by q2 is equivalent to rotation by q2q1.
Geometric intuition
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Comparison of Quaternions, Euler Angles and Matrices Euler Angles/Matrices – Disadvantages
Lack of intuition The order of rotation axes is important. Gimbal lock
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. Mathematically gimbal lock corresponds to loosing a degree of
freedom in the general rotation matrix.
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Comparison of Quaternions, Euler Angles and Matrices Euler Angles/Matrices – Disadvantages
Gimbal lock
If we letβ=π/2, then a rotation with αwill have the same effect as applying the same rotation with -γ.
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.
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Comparison of Quaternions, Euler Angles and Matrices Euler Angles/Matrices – Disadvantages
Implementing interpolation is difficult Ambiguous correspondence to rotations The result of composition is not apparent The representation is redundant
Euler Angles/Matrices – Advantages The mathematics is well-known and that matrix
applications are relatively easy to implement.
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Comparison of Quaternions, Euler Angles and Matrices Quaternions – Disadvantages
Quaternions only represent rotation Quaternion mathematics appears complicated
Quaternions – Advantages Obvious geometrical interpretation Coordinate system independency Simple interpolation methods Compact representation No gimbal lock Simple composition
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Interpolation of Solid Orientations
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Rigid Body (Single Particle)Newtonian Mechanics (Single Particle)
Newton’s Laws A body remains at rest or in uniform motion
unless acted upon by a force. Force equilibrium
A body acted upon by a force moves in such a manner that the time rate of change of momentum equals the force. 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.
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Rigid Body (Single Particle)Newtonian Mechanics (Single Particle)
Inertial Frame 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. F = d(mv)/dt = m dv/dt = mr’’ Example
Projectile motion in two dimensions.
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Rigid Body (Single Particle)Newtonian Mechanics (Single Particle)
Question No air resistance. Muzzle velocity of the projectile: v0. Angle of elevation Θ. Calculate the projectile’s displacement,
velocity and range.
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Rigid Body (Single Particle)Newtonian Mechanics (Single Particle)
Solution.
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Rigid Body (Single Particle)Newtonian Mechanics (Single Particle)
Conservation Theorems The total linear momentum P of a particle is
conserved when the total force on it is zero. P·s = constant
The angular momentum of a particle subject to no torque is conserved. L = r X P.
The total energy E of a particle in a conservative force field is a constant in time. E = T(kinetic) + U(potential)
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Rigid Body (Single Particle)Newtonian Mechanics (Single Particle)
Example 2: A cylinder on a curved surface A cylinder of mass m and radius R1 rolling
without slippage on a curved surface of radius R.
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Rigid Body (Single Particle)Newtonian Mechanics (Single Particle)
Solution
dE/dt = dT/dt + dU/dt = 0
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Geometry of Deformable Models
The models are 3D solids in space.
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Global Deformations
The reference shape s is defined as
T: global deformation e: geometric primitive defined
parametrically in u and parametrized by the variables ai.
The vector of global deformation parameters
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Local Deformations
The displacement d anywhere within a deformable model is represented as a linear combination of an infinite number of basis functions bj(u)
The diagonal matrix Si is formed form the basis functions and where qd,I are local degrees of freedom.
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Kinematics and Dynamics
Kinematic and dynamic formulation of the deformable models Kinematic formulation: the computation of a
Jacobian matrix L It allows the transformation of 3D vectors into q-
dimensional vectors. Dynamic formulation: based on Lagrangian
dynamics and generalized coordinates.
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Kinematics
The velocity of a point on the model
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Kinematics
Computation of R and B using Quaternions
The dual matrix of the position vector p(u) = (p1,p2,p3)T
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Dynamics
Lagrange Equations of Motion
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Dynamics
Kinetic Energy: Mass Matrix
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Dynamics
Acceleration and Inertial Forces
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Dynamics
Acceleration and Inertial Forces
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Dynamics
Damping Matrix: Energy Dissipation
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Dynamics
Stiffness Matrix: Strain Energy
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Dynamics
Stiffness Matrix: Strain Energy
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Dynamics
External Forces