Time-dependent Shapes (I)

1
Time-dependent Shapes (I)
2
Introduction

• In physical simulation collisions occur when two
  objects move through space and hit one another
• Whenever two objects attempt to inter-penetrate
  each other, it is called as a collision. The
  related issue is response to collisions once they
  are detected
• A straightforward method of collision detection
  involves the solution of a sequence of static
  problems one per time step
• ballmovie2.avi

3
What is a Motion?
4
Definition of an Objects Configuration

• The configuration of an object is a
  specification of the positions of all the points
  in this object relative to a fixed coordinate
  system
• Usually it is expressed as a vector of
  position and orientation parameters

5
Rigid Object Example
Objects configuration is q (x,y,q)
In a 3-D workspace q would be of the form
(x,y,z,a,b,g)
6
Disc Robot in 2-D Workspace
7
C-Obstacle for Articulated Robot
8
Collision detection

• Illustration of a time-dependent shape

9
Collision detection for particles

• Consideration of a problem of nonoverlapping
  arrangements of D-dimensional spheres, D 1,2,3,
  confined to a rectangular region ?d of size
  L1xL2x...xLD leads to search of an interaction
  time of particles
• Let the center positions of the N spheres in ?d
  be denoted by the set of vectors r1...rN.
• For two spheres with the diameters a1 and a2,
  interaction time t is the least positive real
  solution of the equation
• At2 2Bt C 0,
• which is derived from r vt2 (a0t)2, where a0
  a1 a2

10
Collision detection for polygons

• A correct test must consider edges and triangles
• However, in many cases merely testing points
  versus triangles produces acceptable results
• Polygon-polygon collision detection is performed
  by testing for the penetration of each vertex
  point of one polygon through the plane of the
  other and by checking that no two edges intersect

11
Collision detection for polygons

• Figure illustrates path point and triangle
  intersection
• This technique is used to detect collision
  between flexible surfaces. Two cases are
  considered depending on whether the surface
  polygon is fixed or moving

12
Collision detection for polygons

• When the surface polygon is fixed a triangle
  with vertices (P0,P1,P2), the parametric vector
  equation is given by
• P (P - P)t P0 (P1 - P0)u (P2 - P0)v,
• where P and P are the beginning and ending
  position of the point, u and v are the parametric
  variables for the plane defined by the triangle
  and t is a time variable for the simulation step
• By solving this system for t, u,v one determines
  whether the point has intersected the triangle
  during the time step by checking the conditions 0
  ? t ? 1, u ? 0, and u v ? 1

13
Collision detection for polygons

• When the surface polygon is moving, the following
  parametric vector equation is setup
• P V t P0 V0t ((P1 - P0) (V1 - V0)t)u
  ((P2 - P0)
• (V2 - V0)t)v,
• where P is the point with velocity V per time
  step, V0, V1, V2 are the respective velocities of
  the points P0, P1, P2
• If we eliminate u and v from the three component
  equations, we get a fifth order polynomial in t,
  which can be solved numerically. Each value of t
  thus arrived at is used to get values for u and v
  by back substitution, and then the standard 0 ? t
  ? 1, u ? 0, and u v ? 1 conditions are used to
  determine whether a collision has occurred.

14
Collision detection for convex polyhedra

• The two polyhedra are assumed to be convex ones
  concave polyhedra can be decomposed into
  collections of convex ones
• The method for detecting collisions is based on
  the Cyrus-Beck clipping algorithm
• The two-dimensional Cyrus-beck algorithm tells
  whether a point is inside a convex polygon. It
  takes the dot product of each sides outward
  normal vector (n) with a vector from some point
  (v) on the side to the point in question (p)

15
Collision detection for time dependent shapes

• Most algorithms described in the literature deal
  with rigid bodies and are based on some kind of
  hierarchical representations. The alternative
  approach is based on the idea of sensor
  particles
• Collision detection based on a particles paradigm
  between complex geometric objects represented by
  polygonal models and undergoing rigid motions is
  discussed
• Quite interesting that almost the same approach
  can be used in geometry modeling mesh
  generation and inhancement

16
Examples of particles distribution

• (a) The model of Rocker Arm with randomly
  distributed particles
• (b) Final distribution of particles on the model
  of Rocker Arm
• (c) The Dragon model with randomly distributed
  particles
• (d) Final distribution of particles on the
  Dragon model

17
Sensor particles

• Sensor particles are interacting particles
  distributed on a surface
• Two types of particles that interact in a special
  way are used for determining the minimum
  distance between two models

18
Assumptions

• Each model has a surface that is for the most
  part plain
• Face's adjacency information is available, i.e.,
  there is a list of neighbor faces for each model
  point, or it can easily be created
• Adjacency information does not change during
  deformation, which means that bodies do not
  change their topology
• The model remains plain for the most part and the
  number of convex parts does not increase greatly
  during deformation

19
Particle Interaction

• Radial attractive and repulsive forces act on
  particles
• Unlike the law for electric charges, the laws for
  repulsive and attractive forces can be different
• Reff is the effective radius of attraction

20
Particle Interaction (cont)

• A) Initial position
• B) After one simulation step
• C) After one interaction step

21
Particles Movement

• In the simplest case, particles can be located
  only at vertices. In this case, we can easily
  calculate the potential of forces in every
  neighboring vertex and move a particle to the
  vertex where the potential has the minimum value
• This technique can be generalized for cases when
  the particle can be located on edges and faces

22
Collision Query

• At every collision query we turn on an
  interaction between particles and wait for the
  particle system to relax
• Two numbers, Nmax and Nmin, are used to control a
  particle system to reach almost relaxed state.
  Nmax is the maximum total number of interaction
  steps and Nmin is the minimum number of moving
  particles. Therefore, we wait until either the
  number of particles moved in interaction step is
  less then Nmin or the number of interaction steps
  becomes equal to Nmax

23
Collision Query (cont)

• The values of Nmin and Nmax depend on the model's
  geometry and number of particles n. Nmax should
  be approximately equal to the number of vertices
  between particles placed on the model.
  Experiments show that about 10 of particles, in
  average, are placed near equilibrium position. By
  this means Nmin should be about 10 times smaller
  then number of particles n

24
Collision Query (cont)

• The complexity of performing collision query is
  O(n2). Since the number of particles n is
  essentially smaller than the number of polygons,
  this complexity allows us to perform fast
  collision queries even for models with a large
  number of polygons
• Performance can also be noticeably improved, by
  using caching on the initial steps. If positional
  relationship changes slowly, the influence of
  distant particles remains almost constant. Thus
  we can efficiently cache this influence

25
Results

• Illustration of particle distribution.
• Benchmarks. Collision detection between two
  polygonal models of letters "W", defined by 11920
  polygons, with 5 particles on each model. The
  average time needed to perform a collision
  detection query on active interval (queries nos.
  950-1984) was 12.67 ms., and the maximum time was
  47 ms. average time during the "warming" steps
  (queries nos. 1-49) was 12.06 ms., maximum time
  was 47 ms. also

26
Results (cont)

• Illustration of collision detection between two
  models undergoing deformations
• collision_and_csrbf1.mpg

27
Motion of Rigid Bodies Under External Forces and
Torques

• The general motion of a rigid body can be
  decomposed
• into a linear motion of a point mass equal to
  that of the body located at the center of mass of
  the body under an external force and
• a rotational motion about the center of mass
  under an external torque.
• scrunchies_2160_comp.avi

28
Motion of Rigid Bodies Under External Forces and
Torques

• The linear motion under a force F can be
  calculated by solving the set of coupled
  differential equations
• V? F/m,
• X? V,
• where m is the mass of the object, X is the
  position
• vector, and V is the linear velocity

29
Motion of Rigid Bodies Under External Forces and
Torques

• Euler Equations
• If we choose principal axes as the body fixed
  axes we get a set of simplified coupled first
  order differential equations for the angular
  velocity W, known as the Euler equations
• Ix W?x (Iz -Iy)WzWy Nx
• Iy W?y (Ix -Iz)WxWz Ny
• Iz W?z (Iy -Ix)WyWx Nz
• The x,y,z are the directions of the principal
  axes, I is the moment of inertia tensor of the
  body. It is important to realize that for any
  arbitrary shaped objects, one can find principal
  axes and the principal moments of inertia.

30
Motion of Rigid Bodies Under External Forces and
Torques

• Euler Equations
• These along with
• T?x Wx
• T?y Wy
• T?z Wz
• where T?s are the orientations in the
  principal axes coordinate, form a set which can
  be solved by a numerical technique such as
  Rung-Kutta method.

31
Motion of Rigid Bodies Under External Forces and
Torques

• Impact Dynamics
• When two bodies collide and the contact area of
  one of the bodies is locally planar, one can
  define the normal N to the tangent surface
• Calculate the linear momentum p mv
• And the torque ? r?F, where F is the force
  acting on the centre of mass

32
Penalty-based Multibody Animation

• In penalty-based multibody animation, a
  spring-damper system is used for penalizing
  penetrations.
• A spring-damper system behaves like a harmonic
  oscilator in classical mechanics.
• Spring-damper models are even found in
  biomechanical muscle models.

33
Penalty-based Multibody Animation

• The basics
• The motion of a single rigid body is described in
  the Newton-Euler equations.
• The center of mass is given by r, the orientation
  is given by the quaternion q and the linear and
  angular velocities are given by v and ?
• One needs to find the values of the force F and
  torque ? at a given instant in time in order to
  perform the numerical integration.

34
Penalty-based Multibody Animation

• The basics
• In order to compute, the contact forces springs
  are inserted at all penetrations as shown in
  Figure
• The springs will try to remove the penetrations
• The springs penalize penetrations

35
Penalty-based Multibody Animation

• The basics
• The contact force from each point of contact is
  thus computed as a spring force.
• Call the exernal contributions fext and ?ext and
  the ith spring force fspring and ?spring .
• Now the total force an torque are computed as
• In case of gravity being the only external force,
  we have fext mg, ?ext 0.

36
Penalty-based Multibody Animation

• The basics
• The simulation loop of the penalty method can be
  summarized as
• Detect contact points
• Compute and accumulate spring forces
• Integrate equations of motion forward in time
• There are, however, several difficulties
  associated with it

37
Penalty-based Multibody Animation

• The basics
• First of all it is not trivial to compute
  satisfactory contact normals or penetrating
  depths (lack of global knowledje).

38
Penalty-based Multibody Animation

• The basics
• Some contact points like the face-face case
  appearing in a box stack is difficult to handle.
• To alleviate this problem, researches have tried
  to sample the entire contact region with contact
  points.

39
Penalty-based Multibody Animation

• Modeling Friction
• Friction is very imporatnt to seem physically
  plausible fo r a human observer.
• It seems reasonable to use a spring-damper system
  for friction force.
• At first point of contact, an anchor point, a, is
  used.
• The friction is determined by the spring force
  stemming from the zero-length spring with spring
  coefficient k between the anchor point and the
  current position p of the contact point.
• Ffriction k(a p).

40
Penalty-based Multibody Animation

• Modeling Friction
• The general idea is illustrated in Figure
• During the simulation we keep an eye on whether
  the following condition holds
• ?Ffriction ? ? ? ? Fnormal?.
• Here ? is the coefficient of friction. Fnormal is
  the normal force.
• If the condition is fulfilled we have static
  friction and we do nothing.
• On the other hand we have dynamic friction.