EECS 366/466
Computer Graphics Project
Spring 2004
Physical Based
Modeling for Simulation
by Ozkan Bebek
Introduction:
Physically based modeling is an important approach to computer animation and
computer graphics modeling. It can be considered as the combination of physics
(control) and computer graphics. As the computation power increase the
animations became more and more realistic. As a part of the realization of the
real world to the screen, physical based modeling has an important role. It
helps us to visualize the objects when they are interacting with the
environment or any controls are applied. In this project physical based
modeling of rigid bodies will be implemented. Using different objects,
interaction of the objects will be observed.
Background:
In this project C++ and OpenGL was used as the computer graphics tool. First
the dynamical modeling of the objects, interactions between ground and objects
was realized. Discrete time dynamical simulation was implemented by solving
dynamical equations of the systems in every step time, T=0.1 ms. The outputs of
the models were animated to the screen in every 10 steps to make system work
closer to real time.
Rigid Body Dynamics
Lets
define the state vector
for a rigid body where x(t) gives the objects
position in the world coordinates and R(t) is the rotation matrix that
includes the orientation information of the body at time t. P(t)
and L(t) are linear and angular momentums. The goal in rigid body
dynamic simulations is to find the changes in the states of an object by using
dynamical equalities under the effect of external forces applied. Sum of all
the forces acting on the particle: gravity, wind, spring forces, etc can
defined as function F(t). If the object has mass m, then the
change of X over time is given by:
To describe a rigid body in space we
need to know its orientation. In addition to the translation of the body in
space, x(t), a rotation matrix, R(t) defining the orientation of
the body is used. So if an arbitrary point in the body is described with p0,
then the position of the point in world coordinates is:
p(t)=
R(t)p0+x(t)
When external forces are applied to
the rigid body, Fe(t), as well as the torque τe(t),
applied can be described as:
Where r(t) is the an arbitrary
point on the rigid body that force is applied.
Linear and angular momentums of a
rigid body can be defined as:
So, for a
rigid body, defined state vector X(t) is shown above. Thus, the state of
a rigid body is its position and orientation (describing spatial information),
and its linear and angular momentum (describing velocity information).
Computation of the state vector for each time step will give the necessary
information to find the values of the state variables for the next step time.
Simulation:
For every object necessary state variables are defined. Thus every object is
physically modeled. At any time auxiliary quantities, variables other than
states, could be calculated and used for animation purposes.
v(t)=P(t)/m I(t)=R(t)IbR(t)T w(t)=I(t)-1L(t).
The
constant quantities are calculated before starting the simulation, i.e. mass, Ib,
Ib-1. After the initialization of the variables the
simulation loop is as follows:
0.
Initialize the variables; calculate necessary variables before main simulation
loop.
1.
Form the state vector by using the quantities available, Xk+1
2.
Copy the variables to current state Xk
3.
Form the 
using the auxiliary quantities.
4.
Using an ODE solver, solve for every
time step, T, next state Xk+1.
E.g.: Euler Forward iteration: 
5.
Derive the auxiliary quantities using Xk+1.
6.
Display the objects using the states.
7.
Return Step 1.
The
describe algorithm executed in an inner loop under the main algorithm. Selected
step time for simulation was 0.1ms. Objects are displayed for every 10
simulation cycles. External forces, such as friction, normal forces are
calculated for every step. A Collision detection algorithm used and contact points
are modeled as a spring-damper systems.
Results
and Conclusion:
Physical
Based Modeling for different objects, Crate, Beach Ball was implemented. Using
OpenGL as a tool, simulation results are displayed. Simulation provides
satisfying performance close to real time. The bouncing and rolling of the
objects are very life-like.
The
mass-spring systems for the contacts are tuned for better visualization. After
some simulation time Euler integrator becomes very unstable, causing
unexplained rolling and jumping of the objects. So using higher order ODE
solvers is more appropriate for the simulation.
References:
Spong, M.,
M. Vidyasagar, ”Robot Dynamics and Control”, John Wiley and Sons, 1989.
Luh,
Walker, Paul; On-line Computational Scheme for Mechanical Manipulators, Journal
of Dynamic Systems, Measurement, and Control; pp 69-76; June 1980.
http://www.ioi.dk/Homepages/thomasj/
http://www.pixar.com/
http://nehe.gamedev.net/