Chap 7 Physically Based Animation

1
Chap 7Physically Based Animation
2
Physically Based Animation

• Forces are typically used to maintain
  relationships among geometric elements
• Accuracy vs. physical realism
• Animators is not necessarily concerned with being
  accurate but rather with believabibity
• Some forces may not relate to physics at all
• They may model constraints that an animator may
  wish to enforce on the motion

3
Physically Based Animation

• When modeling motion using physics principles, we
  need to decide the level at which to model the
  process
• Procedure
• Computationally less expensive, easier to program
• Lacks flexibility
• Purely physics
• Computationally expensive
• More flexible

4
Physically Based Animation

• When physical models are used, animator
• is relieved of lower level specification of
  motions
• Needs to be concerned with specifying high-level
  relationships or qualities of the motion

5
Outline

• Basic physics
• Spring meshes
• Particle systems
• Rigid body simulation
• Enforcing soft and hard constraints
• A Good reference
• Read Witkin and Baraffs SIGGRAPH01 course
  notes Physics-based modeling at
  http//www.pixar.com/companyinfo/research/pbm2001/
  index.html

6
Basic physics

• Newtons laws of motions

7
Basic physics
8
Basic physics
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Spring meshes

• Flexible objects
• Flexibility is modeled by a spring-mass-damper
  simulation simulating the reaction of the body to
  external forces.
• Each vertex a point mass
• Assigned by animators
• Evenly distributed among objects vertices
• Each edge a spring
• Rest length is original edge length
• Spring constants
• Arbitrary, assigned uniformly throughout the
  object

10
Spring meshes

• Spring-mass-damper simulation
• As external forces are applied to specific
  vertices
• Vertices will be displaced relative to other
  vertices
• The displacement induces spring forces, which
  will impart forces to the adjacent vertices and
  reactive forces back to initial vertices.
• Force propagation one time step at a time

11
Spring meshesA simple example
Instant force F applied to V2. Length of springs
E12 and E23 changes. Spring forces impart
restoring forces to V1 and V2, and V2 and
V3 Since external force is wrongly assumed
constant throughout the time step, a spring
simulation may numerically explode (the motions
get larger and larger). A smaller time step, or
smaller spring constant, or larger masses can be
used but slow down the simulation. A common
method add dampers.
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Spring meshesDampers

• A damper introduces an additional force in the
  spring that works against the velocity of the
  spring length, thus helping to limit the speed at
  which a spring changes length.
• Help to control the change in length and keep it
  within some range that does not allow the
  simulation to explode.

13
Spring meshes

• Modeling with only object edges with spring
  dampers can result in a model that has more than
  one stable configuration.
• Additional spring dampers can help to stabilize
  the shape.

If a cubes edges are modeled with spring
dampers, during applications of extreme
external force is applied, the cube can turn
Inside out.
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Spring meshesAngular spring dampers

• angular spring resists deviation to the rest
  angle between faces and imparts a torque along
  the edge that attempts to restore the rest angle.

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Spring meshesVirtual springs

• Virtual springs introduce forces into the system
  that do not directly model physical elements. For
  example,
• Example on the last page
• Virtual springs with zero rest lengths can be
  used to constrained one object to lie on the
  other
• Virtual springs with non-zero rest lengths can be
  used to maintain separation between moving
  objects.

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Particle systems

• Particles have been used to animate many
  behaviors, such as gases, water, fire, rubber,
  clothes, flocking, hair.

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Particle systemsAssumptions

• Particles do not collide with other particles
• Particles do not cast shadow on each other,
  except in an aggregate sense
• Particles only cast shadow on the rest of
  environment
• Particles do not reflect light
• Particles often have a finite life span

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Particle systemsSteps in a frame

• Generate any new particles
• Each new particle is assigned attributes
• Any particles that have exceeded their life span
  are terminated
• The remaining particles are animated and their
  shading parameters changes according to the
  controlling process
• The particles are rendered

19
Particle generation

• For each frame, particles are generated according
  to a controlled stochastic process
• User-specified distribution centered at the
  desired average number of particle per frame
• Can be function of time

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Particle attributes

• Particle properties
• mass
• position
• velocity
• force accumulator
• age, lifespan
• rendering properties (shape parameters, color,
  transparency)
• Each of the attributes is initialized when the
  particle is created.

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Particle life span
At each new frame, each particles lifetime is
decremented by one When the attribute reaches
zero, the particle is removed from the system.
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Particle animation

• Each particle is animated throughout its life.
• Includes position, velocity, and rendering
  attributes
• Position and velocity
• Users consider forces and compute the resultant
  particle acceleration, update its velocity, and
  update the position.
• Color and transparency can be a function of time,
  its own life span remaining, its height, and so
  on. Shape can be a function of its velocity

23
Particle animation Forces on particles

• Forces can be unary, particle pair, or
  environmental.
• Unary forces
• Gravity, viscous drag
• Particle pair forces
• Can be represented by springs and dampers, if
  desired.
• Environmental forces
• Arise from a particles relationship to the
  environment

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Forces on particles

• Particles respond to forces
• We represent this using differential equations

2nd order ODE
1st order ODEs
25
Unary Forces

• Forces that only depend on one particle

fdrag
v
Viscous Drag
Gravity
Wind Fields
f mg
f kvwind
f -kdv
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n-ary Forces

• Forces that depend on n particles
• Example binary forces between two particles -
  spring and damper

Springs
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n-ary Forces Spring Force

• If particle is located farther than the rest
  position, the spring force needs to pull it back
• If the particle is located nearer than the rest
  position, the spring force needs to push it away
• Combine two cases

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n-ary Forces Damping Forces

• According to the law of energy conservation, a
  particle system consists of only masses and
  springs keep bouncing from each other after
  external forces disappear
• Damping/viscous drag force resist motion, making
  a particle system gradually come to rest in the
  absence of external forces

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n-ary Forces Damping Forces (cont.)

• It is highly recommended that at least a small
  amount of damping is applied to each particle
• Excessive damping, however, makes a particle
  appear that floating in molasses (energy
  dissipates out too quickly, not responsive)

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n-ary Forces Damper Force

• If two particles are departing, the damper force
  needs to pull them back
• If two particles are approaching, the damper
  force needs to push them away
• Combine two cases

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Spatial Forces

• Forces that depend on nearby particles within a
  local region

Gravity, Lennard-Jones and electric potentials
32
Particle rendering

• Several methods
• Model each particle as a point light source
• Each particle is rendered to a small graphical
  primitive (blob).
• Particles that map to the same pixels in the
  image are additive - the color of a pixel is
  simply the sum of the color values of all the
  particles that map to it
• Model each particle as a textured billboard
• Polygon facing the viewer, texture
• Rendered as Metaballs in off-line rendering
  isosurfaces computed from particle-metaballs make
  quite convincing liquids.

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Particle renderingParticle
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Particle renderingParticle
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Particle rendering Billboard texture
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Particle rendering Billboard texture
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Particle rendering Billboard texture
From Mark Harriss work http//www.markmark.net/cl
ouds/
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Particle renderingBillboard texture

• An impostor replaces a cloud with a billboard
  textured with an image of the cloud from a
  certain viewpoint.
• Image is updated only when translation of the
  viewpoint introduces enough error in the image
• Impostors can be reused for many frames.
• By using impostors, we are able to render cloudy
  scenes of hundreds of clouds and hundreds of
  thousands of particles at very high frame rates.

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Particle renderingBillboard texture
Figure 6 Impostor generation and translation
error metric.
Impostor generation and translation error metric
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Particle rendering
Video Cloud rendering by Niniane
Wang http//www.ofb.net/niniane/clouds/
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Particle renderingMetaballs and isosurface
From Screen Space Fluid Rendering with Curvature
Flow http//industrialarithmetic.blogspot.com/20
09/01/ our-paper-screen-space-fluid-rendering.html
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Collision Detection

• Determine when a particle has collided an object
• Particle has collided if and only if

Objects surface
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Collision Response

• What should we do when a particle has collided?
• The correct thing to do is rollback the
  simulation to the exact point of contact
• Easier to just modify positions and velocities

After the collision
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Contact Forces

• When the particle is on the collision surface a
  contact force resists penetration
• Contact forces do not resist leaving the surface
• Simple friction can be modeled

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Structure of Particle Systems

• Separate the data structures and integration

state
Particle System
Solver
send data as 6n vectors
time
particles
state
derivatives
state/derivatives
Particle
x v f m
x v f m
x v f m
x v f m
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Structure of Particle SystemsImplementation

• Solver Interface

Solver
System
GetDim
6
Get/Set State
x v f m
x v
v f/m
Deriv Eval
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Structure of Particle SystemsImplementation

• Solver Interface

Solver
System
GetDim
6n
particles
Get/Set State
x1 v1
x2 v2
xn vn
n
time
v1 f1/m1
v2 f2/m2
vn fn/mn
Deriv Eval
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Physics Engines for Dynamics Simulation

• Open Dynamics Engine (ODE)
• Free software http//www.ode.org/
• high performance library for simulating rigid
  body dynamics.
• It is fully featured, stable, mature and platform
  independent with an easy to use C/C API.
• It has advanced joint types and integrated
  collision detection with friction.
• ODE is useful for simulating vehicles, objects in
  virtual reality environments and virtual
  creatures.

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Physics Engines for Dynamics Simulation

• Havok
• Commercial software
• Reactor in 3ds MAX
• State-of-the-art game physics solution, for use
  with in-house game animation systems
• Havok Destruction is the cross-platform tool for
  simulation of rigid body destruction.

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Physics Engines for Dynamics Simulation

• NVIDIA PhysX
• A powerful physics engine which enables real-time
  physics in leading edge PC and console games.
• Is widely adopted by over 150 games, is used by
  more than 10,000 registered users.
• Designed specifically for hardware acceleration
  by powerful processors with hundreds of cores.
  Combined with the tremendous parallel processing
  capability of the GPU, PhysX will provide an
  exponential increase in physics processing power

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Physics Engines for Dynamics Simulation

• AGEIA PhysX accelerator
• The worlds first dedicated physics processor
  designed to support extreme physical gaming
  interactions.
• Its highly parallel, interactive PhysX cores are
  optimized for dynamic, large-scale, physics
  processing to accelerate physical motion and
  interaction at a scale and quality far beyond
  that of general purpose processors.

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Rigid Body Simulation

• Various forces to be simulated are modeled.
• When the forces are applied to objects, they
  induce
• linear acceleration (based on objects mass)
• angular acceleration (based on masss
  distribution)
• These accelerations are integrated over a delta
  time step to get changes of objects velocities,
  which in turns integrated over a delta time step
  to produce changes in position and orientation.

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Rigid Body Simulation

• Various forces to be simulated are modeled.
• When the forces are applied to objects, they
  induce
• linear acceleration (based on objects mass)
• angular acceleration (based on masss
  distribution)
• Read Witkin and Baraffs SIGGRAPH01 course
  notes Physics-based modeling
• http//www.pixar.com/companyinfo/research/pbm2001/
  index.html

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Rigid Body SimulationUpdate cycle
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Rigid Body Simulation

• Continuous process vs. discrete time simulation
• Assume acceleration is constant over the delta
  time step
• These accelerations are integrated over a delta
  time step to get changes of objects velocities,
  which in turns integrated over a delta time step
  to produce changes in position and orientation.
• How to update?
• Euler integration method
• Runge Kutta method Second-order in magnitude
  of error term

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Physics-based Simulation

• A procedure that generates a sequence of the
  states of a system based on physics laws

xi
Newtonian laws elastic forces wind gravity frictio
n
xi
xi1
xi1 xi ?x
?x
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Differential Equations

• Differential equation describes the relation
  between an unknown function and its derivatives
• Solving a differential equation is to find a
  function that satisfies the relation
• Numerical solution of differential equations is
  based on finite-dimensional approximation

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Ordinary Differential Equations

• Ordinary differential equation (ODE)
• All derivatives are with respect to single
  independent variable, usually representing time

Known function
Unknown function that evaluates the state given
time
Time derivative of the unknown function
Well show that a higher ODE can be transformed
into this 1st order system soon!
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Higher-Order ODEs

• Order of ODE determined by highest-order
  derivative of solution function appearing in ODE
• Equations with higher derivatives can be
  transformed into equivalent first-order system

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Higher-Order ODEs (cont.)

• Given k-th order ODE
• Define

• Original ODE equivalent to first order system

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Visualizing Solution of ODE
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Vector Field

• The complete set of all solutions to the
  differential equation
  defines a vector field over (x, t) plane.

Think of this vector field as the sea, and the
velocity of current at different places and time
is defined by f(x,t)
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Initial Value Problem

• For a particular initial value, the solution is a
  curve.
• Given the starting point, follow the integral
  curve

Release a ball at any starting point and let it
drift following the current. The trajectory swept
out by the ball is an integral curve.
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Numerical Solution of ODEs

• Instead of true integral curve, numerical
  solution follow a polygonal path
• Each leg is obtained by evaluating the derivative
  at discrete time steps
• Bigger steps, bigger errors

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Issue I Inaccuracy

• Error turns x(t) from a circle into the spiral of
  your choice!
• May jump to other circles

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Issue II Instability

• May diverge!

to Neptune!
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Eulers Method

• Simplest numerical solution method
• Bigger time steps, bigger errors

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Eulers Method (cont.)

• Solves ODE using one-term Taylor-series

O(h2) 2nd order accurate
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Eulers Method (cont.)

• Truncation error in s single step O(h2)
• The error in a complete solution is an
  accumulation of all the single step errors

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Rigid Body Simulation Euler integration method
(Book)

• Explicit Euler integration method

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Drawbacks of Eulers Method

• Too inaccurate to be used in practice
• Inefficiency
• Need to use small time-steps to avoid divergence
• Example http//www.cse.uiuc.edu/iem/ode/eulrmthd/
  
• Improvement using the midpoint method
• Slope at midpoint is used

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Runge-Kutta methodThe Midpoint Method

• a. Compute an Euler step
• b. Evaluate f at the midpoint
• c. Take a step using the midpoint value

Assume f depends on t only indirectly through x
x2
x1
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Runge-Kutta method The Midpoint Method (cont.)

• Solves ODE using two-term Taylor-series
• Approximating f by Taylor-series

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Runge-Kutta method The Midpoint Method (cont.)
Turning O(h2) to O(h3)
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Runge-Kutta method The Midpoint Method (Book)

• Runge Kutta method
• Symmetric w.r.t. the time interval
• Second-order in magnitude of error term

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Runge-Kutta 4th Order Method

• Using a weighted average of slopes obtained at
  four points

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Runge-Kutta 4th Order Method (Book)

• Fourth-order Runge Kutta method

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Adaptive Step Size

• Ideally, we want to choose h as large as
  possible, but not so large as to cause big error
  or instability
• We can vary h as we march forward in time

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Adaptive Step Size for Eulers Method

• Estimate xa by taking a full Euler step
• Estimate xb by taking two half Euler steps
• xa and xb differ from the true value of x(tah)
  by O(h2)
• e xa - xb is bounded by O(h2), giving us
  convenient estimate to the error in taking an
  Euler step of size h
• How to do adaptive steping ?

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Adaptive Step Size for Eulers Method

• What is optimal h if we want to have an error of
  as much as e, and the current error is e?

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Take Home Message

• Dont use Eulers method
• Inaccuracy
• Inefficiency (or unstable)
• Do use adaptive step size
• Read Witkin and Baraffs SIGGRAPH01 course
  notes Physics-based modeling
• http//www.pixar.com/companyinfo/research/pbm2001/
  index.html

83
Motion equation for a rigid body

• To develop the equations of motion for a rigid
  body, we need to discuss
• Linear force vs. rotational force
• Torque
• Linear momentum vs. angular momentum
• Inertia tensor
• Equations

84
Orientation and rotational movement angular
velocity

• Angular velocity w(t)
• Direction Axis of rotation,
• Speed of orientation in revolutions per unit of
  time
• For linear motion, x(t) and v(t) are related by
• How are R(t) and w(t) related?
• We need to find how the derivative of a vector in
  a rigid body is related to w(t)

85
Orientation and rotational movement angular
velocity of a particle

• Angular velocity of a point

p
86
Orientation and rotational movement angular
velocity of a particle
87
Orientation and rotational movement angular
velocity of a particle
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Orientation and rotational movement angular
velocity of a rigid body

• Angular velocity of a rigid body
• Position and orientation of a rigid body are
  represented by x(t), and R(t)
• x(t), and R(t) are used to transform the
  body-space description into world space

89
Orientation and rotational movement angular
velocity of a rigid body

• R(t) specifies a rotation of the body about the
  center of mass
• It is a matrix of 3 orthogonal column vectors
• At time t,
• represents the x-axis of body space in world
  space
• Each column vector represents a vector from the
  origin
• A point Q on the rigid body with local coordinate
  q

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Orientation and rotational movement angular
velocity of a rigid body
91
Orientation and rotational movement angular
velocity of a rigid body

• At time t, the derivative of the first column of
  R(t) is the rate of change of this vector. So

92
Orientation and rotational movement angular
velocity of a rigid body

• A point Q on the rigid body with body-space
  coordinate q.
• Then the world-space location of q(t) is the
  result of first rotating q about the origin and
  then translating it

This separates the velocity of a point on a
rigid body into two components A linear
component v(t) A angular component w x (q(t)-x(t))
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Orientation and rotational movementCenter of mass

• Integration of the differential mass times its
  global position in the object

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Orientation and rotational movementForces and
torque

• Linear force vs. rotational force

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Orientation and rotational movementLinear
momentum

• Linear momentum p of a particle with mass m and
  velocity v
• Linear momentum of a rigid body

Total linear momentum of a rigid body is the same
as if the body is simply a particle with mass m
and velocity v(t).
96
Orientation and rotational movement Linear
momentum

• Derivative of linear momentum and force
• Instead of v(t), we use P(t) as a state variable
  for the rigid body.
• Be more consistent with the way we deal with
  angular velocity and acceleration.

97
Orientation and rotational movementAngular
momentum (book)_

• Angular momentum of a particle

98
Orientation and rotational movement Angular
momentum (book)

• Angular momentum of an object

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Orientation and rotational movementAngular
momentum

• Angular momentum of a rigid body
• Similar to linear momentum
• where I(t) (called inertia tensor) is a 3x3
  matrix that describe how the mass in the body
  is distributed relative to the bodys center of
  mass.
• I(t) depends on orientation of the body but does
  not depend on the bodys translation

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Orientation and rotational movementInertia tensor

• At a given time t, let be the
  displacement of particle i from bodys mass
  center x(t) by
• .
• I(t) is expressed in terms of ri as the
  symmetric matrix

101
Orientation and rotational movementInertia tensor

• How to compute I(t) for a rotated dody?
• Compute at run-time? Too expensive!
• Pre-computed I(t) in body-space coordinates and
  then compute it at run time after rotation.

102
Orientation and rotational movementInertia tensor
103
Orientation and rotational movementInertia tensor
104
Orientation and rotational movementInertia tensor
Constant over the simulation
Let
We have
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Orientation and rotational movementInertia tensor
106
Orientation and rotational movementInertia tensor

• Linear momentum and linear velocity
• Angular momentum and angular velocity
• I(t) inertia tensor, describes how the mass is
  distributed about the center of mass in space
• For an un-transformed object Iobject
• For a transformed object (depends only on
  orientation)

107
Motion Equation

• The state of a rigid body
• Spatial information
• Position
• Orientation
• Velocity information
• Linear momentum
• Angular momentum
• Known information
• Mass
• Body-space inertia

108
Motion Equation
109
Bodies in collision

• Objects moving in a dynamic scene
• Objects in collision need to do the detection
• Objects sliding against on other object
• Objects resting on each other
• Need to calculate forces for computing accurate
  reaction

110
Bodies in collision

• Ultimate goal in VR
• Real and virtual objects need to behave like real
  ones.
• At a minimum
• Objects should not pass through each other, and
  things should move as expected when pushed,
  pulled, or grasped.
• Overall
• Physical simulation in VR must run reliably,
  seamlessly, automatically, and in real time.

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Collision detectionComputer Animation
112
Collision detectionHaptic Interaction
From J.C. Latombes notes
113
Collision detectionMotion/Path Planning
From J.C. Latombe
From James Kuffner
114
Collision detectionCrowd Simulation
From J.C. Latombes notes
115
Bodies in collision

• To prevent interpenetration
• Collisions must be detected.
• Velocities must be adjusted in response to
  collisions.
• Collision response must be computed.
• If the collision response does not cause the
  objects to separate immediately, contact forces
  must be calculated and applied until separation
  finally occurs.

116
Problems of Collision Detection

• A naive collision detection has
• Fixed-timestamp weakness
• All-pair weakness
• Pair-processing weakness
• Features that current methods have
• Interactive rate
• Handle polygon soaps
• Models can undergo rigid-body motion
• Provide well-fit bounding volume
• Collision detection only occurs at discrete times.

117
Performance of a naive method

• All-pair weakness
• If the scenario contains n moving objects and m
  static objects, total object test for each frame
  is
• Pair processing weakness
• Note
• Performance evaluations for collision detection
  are extremely difficult since algorithms are
  sensitive to the actual scenarios, and there is
  no algorithm that performs best in all cases.

118
Speed Up Methods

• Bounding volume for object level and polygon
  level.
• Pure bounding volume
• Hierarchical bounding volumes
• Support progressive collision detection.
• Spatial coherence
• Usually large regions of the space are occupied
  by only one object or none at all. If some
  objects share the same region, they are likely
  intersecting.
• Time coherence
• Moving objects usually move on a continuous path.
  Results of earlier collision query can be
  exploited.

119
Two-level Collision Detection

• Two-level Collision Detection
• Globally searching pairs of objects that have
  potential collision
• Spatial partition
• Dimension reduction and sorting (Sweep-and-Prune)
• Collision detection on a pair of objects
• Hierarchical CD based on OBBTree, AABBTree,
• Polygon-polygon overlap test

119
120
Two-level CD system
Pruning Multi- Body pairs (Sweep/Prune)
Exact collision Detection (OBB tree)
Simulation
Collision response
121
Space Partition

• Spatial coherence is usually exploited by
  partitioning the space.
• All-pair weakness can be resolved by the use of
  spatial coherence.
• Space partition methods differ in fast
  neighbor-finding (for static environments) and
  fast updating for moving objects.
• Grid (uniform partition)
• Octree
• Binary space partition

122
Temporal Coherence One or Two-Dimension Sweep
and Prune

• Sorting objects in 3-D space by dimension
  reduction
• If two objects collide in a 3D space, their
  orthogonal projections onto the xy, yz, and
  xz-planes and x, y, and z-axes must overlap.
• Project bounding volume of objects.
• Based on Axis-aligned bounding boxes (AABB).
• Fixed-size bounding cube large enough to contain
  the object at any orientation.
• Dynamically-resized rectangular bounding boxes

123
One or Two-Dimension Sweep and Prune

• Project each 3D box onto x, y, and z axes.
• Construct 3 lists, one for each dimension.
• Each list contains the values of the endpoints of
  the projected interval corresponding to that
  dimension.
• Determine which intervals overlap by sorting
  these lists.
• Utilize temporal or frame coherence by changing
  only the values of the interval endpoints.
• Bubble sort or insertion sort work well for
  previously sorted lists.

124
One or Two-Dimension Sweep and Prune
Implementation issues - -1

• Fixed-sized cubes are preferred over
  dynamically-resized boxes, except for the class
  of oblong objects.
• Which sorting method is better?
• Bubble sort works better for environments where
  only a few objects move, such as walkthrough
• Insertion sort works better for environments
  where large number of objects move.
• Overlap status consists of a Boolean flag for
  each dimension. Whenever all three of these flags
  are set, the bb of objects pair overlap.

125
Bounding Volumes

• Bounding volume is used to do a pre-check before
  doing any further collision detection.
• Bounding sphere (BS)
  Fast overlapping check, not a
  tight fit.
• Axis-aligned bounding box (AABB)
  Fast overlapping check, a better tight fit.
• Oriented bounding box (OBB) Tight
  fit, but with relatively expensive overlapping
  test.
• K-DOPS (direct oriented polytopes)
  OBB more cutting axes

126
Bounding Volumes
127
Hierarchical Bounding Volumes

• Hierarchical bounding volume is useful
• For efficiently resolving the pair-processing
  weakness, and
• Providing a basis for the hierarchical or
  time-critical scheme for collision detection.
• Three types
• Hierarchical Axis-aligned bounding box
• Hierarchical Oriented bounding box
• Hierarchical Bounding sphere

128
General Hierarchical Collision Detection Scheme

• Common denominators of these schemes
• A hierarchical bounding volume is built for each
  object.
• High-level code for a collision query is similar,
  regardless of the BV type used.
• BV-BV overlapping tests and primitive-primitive
  overlapping tests are different depending on what
  BVs and primitives are used.
• A simple cost function can be used to trim,
  evaluate, and compare performance.

129
Hierarchical Collision Detection Hierarchy
Building

• Common hierarchy used
• K-ary tree, where each node may at most have k
  children.
• At each internal node, there is a BV enclosing
  all its children.
• At each leaf, there are one or more primitives.
• Three ways of building a hierarchy
• Bottom-up
• Incremental tree insertion
• Top-down

130
Hierarchical Collision Detection Bottom-up
Hierarchy Building

• Start
• Starts by combining a number of primitives and
  finding a BV for them.
• Grouping
• This BV is grouped with one or more BVs
  constructed in a similar way, thus yielding a
  new, larger parent BV.
• Recursive grouping
• Repeat the grouping until only one BV exists,
  which becomes the root of the hierarchy.

131
Hierarchical Collision Detection Hierarchy
Building by Incremental tree-insertion

• Start with an empty tree
• All other primitives and their BVs are added one
  at a time to the tree.
• To make an efficient tree, an insertion point
  should be selected so that the total tree volume
  increase is minimized.
• Little is known about this scheme in the context
  of collision detection.

132
Hierarchical Collision Detection Top-Down
Hierarchy Building

• Start
• Find a BV for all primitives of the object, which
  is then acts as the root of the hierarchy.
• Recursively apply a divide-and-conquer strategy
• First to split the BV into k or fewer parts and
  find all included primitives for each such part.
  A BV is then created for each part.
• Potential advantage
• A hierarchy can be created on an as-needed basis,
  i.e., we can construct the hierarchy for those
  parts of the object where it is actually needed.
• Used by majority of hierarchical CD.

133
Hierarchical Collision Detection

• K-ary tree
• K 2 minimizes the work to be done when
  traversing a path from root to the worst-case
  leaf, and it is easier to compute the hierarchy
  than the case of higher k.
• Higher k gives a tree with lower height, but it
  requires more work at each node.

134
Hierarchical Collision Detection

• Challenges
• Find tight-fitting bounding volume.
• Find hierarchy construction that creat balanced
  and efficient tree.
• Efficient BV-BV overlapping test.
• Efficient primitive-primitive overlapping test.
• Notes
• Balanced trees are expected to perform best in
  all cases, since the time of CD query will not
  vary.
• But, it does not mean that it is best for all
  inputs, e.g., those parts that seldom or never be
  queried for a collision can be located deep in
  the hierarchy.

135
Hierarchical Collision DetectionCollision
testing between Hierarchies -1

• Collision detection
• Test whether or not two objects collide, and may
  terminate whenever a pair of triangles has been
  found to be overlapping.
• Collision determination
• In some cases, all pairs of overlapping triangles
  might be wanted.
• Can be solved with small alterations of the code
  for collision detection.

136
Hierarchical Collision DetectionCollision
testing between Hierarchies -2

• FindFirstHitCD(A, B)
• return(TRUE, FALSE)
• if (not-overlap(ABV, BBV) return FALSE
• else if(isLeaf(A)) / try to reduce tested
  pairs /
• if(isLeaf(B))
• for each triangle pair TA in
  AC and TB in BC
• if(overlap(TA, TB))
  return TRUE
• else for each child CB in BC
• 
  FindFirstHitCD(A, CB)
• else for each child CA in AC
• FindFirstHitCD(CA,
  B)
• return FALSE

137
Hierarchical Collision DetectionCost function -1

• Cost function

138
Hierarchical Collision DetectionCost function -2

• Conflicting goals
• Creating a better hierarchical representation
  results in lower values of nv, np, and nu.
• Creating better methods for BV-BV or
  primitive-primitive tests would lower cv, cp.
• Bounding volume examples
• Sphere, AABB, OBB
• K-DOPS (direct oriented polytopes)
• Pie slices
• Spherical shells good fits for Bezier surfaces

139
Hierarchical OBBTree

• Why OBBTree?
• Sphere tree and AABBTree do very well in
  performing rejection tests, whenever two objects
  are far apart but do poorly for objects in close
  proximity and having multiple contacts.
• OBBTree was designed to perform especially well
  if parallel close proximity is found during
  collision detection.
• Parallel close proximity Two surfaces are very
  close and nearly parallel.
• A fast overlapping test for OBB-OBB is provided.

140
Hierarchical OBBTree

• Collision detection based on OBBTree
• Preprocessing
• Building an OBBTree for each object.
• Top-down
• Bottom-up
• Detection computation
• Perform recursively the separating-plane test for
  each pair of objects.

141
Building an OBBTreeTop-down approach

• Split the polygon soap along the longest axis of
  the OBB.
• Pass through the OBBs center
• OBBTree may not balanced.
• If the split fails, the next longest axis can be
  used.
• Pass the point that corresponding to the median
  of the projection of all triangle centroids onto
  this axis.
• OBBTree can be balanced.
• Partition the polygons into two sub-groups based
  on their centroids.
• After split, an OBB is found for each sub-group.

142
Building an OBBTreeTop-down approach
143
Building an OBBTreeDerive a good-fit OBB -1

• For a given triangle soap, compute an orientation
  of the triangles vertices (via the first and
  second order statistics)
• ? average of all vertices
• C the covariance matrix
• Based on C
• The normalized orthogonal eigenvectors of C are
  used as a basis - axes of the OBB.
• External vertices along each axis yields sizes of
  the OBB.

144
Building an OBBTreeDerive a good-fit OBB -2

• OBB for a polygon soap can also be found by
  computing an orientation of the triangles of the
  convex hull
• The OBB will be less dependent on the
  distribution of polygons.
• The convex hull of an object is computed.
• Can be done by, e.g., Quickhull algorithm.
• In return, we have n triangles, and based on
  which a covariance matrix is derived.
• Centroid of the convex hull
• Covariance matrix
• Normalized eigenvectors of C as the OBBs axes.
• Min-max extent on each axis.

145
Building an OBBTreeDerive a good-fit OBB -3
146
Fast Overlap Test for OBBsHandling Rigid-Body
Motions

• In OBBTree, each OBB is stored with a rigid-body
  transformation (a rotation R and a translation
  vector t ) matrix MA.
• When testing CD of OBBs A and B,
• The overlap test should be done in the coordinate
  system of one of OBBs, say A.
• So A is now a AABB.
• Transform B into As coordinate system with

147
Fast Overlap Test for OBBsSeparating Axis Theorem

• A naïve algorithm requires 144 edge-face test
  (12 edges x 6 faces x 2 boxes)
• Separating axis theorem
• For any two arbitrary, convex, disjoint
  polyhedra, A and B, there exists a separating
  axis where the projections of A and B are also
  disjoint.
• Checking for disjoint OBBS involves searching for
  a separating axis.
• If A and B are disjoint, then they can be
  separated by a separating axis that is orthogonal
  to either
• A face of A, or
• A face of B, or
• An edge from each polyhedron.

148
Fast Overlap Test for OBBsSeparating Axis Theorem
Axes p and q are orthogonal to faces of A. Axes s
and t are orthogonal to faces of B. It is
sufficient to find one axis that separates the
projection in order to know that the OBBs do not
overlap. Here, q is the only axis that separates
the projections.
149
Fast Overlap Test for OBBsSeparating Axis Theorem

• General separating-plane test 15 tests
• Three axes of A,
• Three axes of B, then
• Nine axes from A and B (9 axes).
• Sequentially test for each axis candidate
• Do axis projection and form the interval on the
  axis.
• Two objects are disjoint if their intervals don't
  overlap.
• If the intervals overlap, boxes may or may not be
  disjoint. Further tests on other axes may be
  needed.
• More efficient way?
• What are the axes and in which order?

150
Fast Overlap Test for OBBs A fast separation
test -1

• Assume that a potential separating axis is l.
• Radii of the OBBs A and B on the axis l

151
Fast Overlap Test for OBBsA fast separation test
-2

• Done in the coordinate system of A
• Formed by As center and axes
• ac, axes au, av , aw
• B is assumed to be relative to A
• By separating axis theorem
• It is sufficient to find one axis that separates
  A and B to be sure that they are disjoint.
• Fifteen axes have to be tested
• 3 from the faces of A (axes of A au, av , aw)
• 3 from the faces of B (axes of B bu, bv , bw)
• 3x39 from combinations of edges from A and B
• (cij ai x bj, i,ju,v,w)

152
Fast Overlap Test for OBBs A fast separation
test -3
Simplify the expression by working in
the coordinate system of A Case 1
153
Fast Overlap Test for OBBs A fast separation
test -4
Simplify the expression by working in
the coordinate system of A Case 2
154
Fast Overlap Test for OBBs A fast separation
test -5
Case 3
155
Fast Overlap Test for OBBs A fast separation
test -6

• If any of these 15 tests is positive, the OBBs
  are disjoint.
• Number of operations
• Maximum 180, or 240 if transformation of B into
  As coordinate system in included.
• Order of axes
• Has an impact on performance.
• First test three axes of A
• They are orthogonal and thus reject the overlap
  faster, and the simplest tests.
• Then three axes of B, followed by the axes formed
  by axes of A and B.

156
Fast Overlap Test for OBBs Another fast
separation test -7

• Simply skip the last 9 axis tests can speed up
  the overlapping test,
• Geometrically, this is amount to doing two
  AABB-AABB tests.
• But sometimes report two disjoint OBBs as
  overlapping.
• In these cases, the recursion in the OBBTree will
  go deeper than necessary.

157
Two-level CD system

• Globally searching pairs of objects that have
  potential collision.
• Spatial subdivision
• Sweep-and-Prune
• Collision detection on a pair of objects
• Hierarchical CD based on OBBTree, AABBTree,
• Polygon-polygon overlap test

158
Two-level CD system
Pruning Multi- Body pairs (Sweep/Prune)
Exact collision Detection (OBB tree)
Simulation
Collision response
159
Time-critical collision detection

• Time critical computing needs to consider all
  computations involved in each frame update.
• Two major computations
• Rendering and Collision detection
• Time-critical collision detection
• Progressive refinement on the discrete-time
  collision detection using hierarchical
  representations, such as
• Sphere tree, box tree (axis-aligned or oriented).
• Hard to estimate the allocated time.