Rigid Body Simulation
By Cheng-Nan Hsieh and Chia-Ming
Chang
CSCI 520 - Computer Animation & Simulation
December 12, 2006
1.0 Objective
Simulation of rigid body dynamics has been
a field of active research for quite some time. Our goal for this project was
to create a realistic simulation of rigid body dynamics, which is billiard
game. In this case, we have to simulate the collisions between balls and
implement the physics engine for this game. In order to achieve this goal, we build
the whole project based on the ideas and algorithm of SIGGRAPH
There are still many potential applications in rigid body simulation. In different fields, such as movie or game, people still rely on rigid body simulation to create realistic scenes or interactive applications.
2.0 Implementation
Our simulation of rigid body consists of three components. The first one is the whole physics system with physics attributes, which are mass, velocity, linear momentum, angular momentum, inertia tensor, force and torque. The second one is detection of contact collisions, which are ball, table plate, table boundaries, and table holes. Between each of them, we implement different functions to change their physics attributes. The third is rendering and visualization, which includes model construction and texture mapping.
2.1 Rigid body
∙ Introduction
Because the billiard game is all about sphere shape, rigid body itself in this case is easy to define. Although the whole system is simple, we also implement the table boundary, score hole, and table. Also, we use quaternion to represent the rotation and other attributes.
∙ Implementation
We use Euler method for ODE solver to calculate position, quaternion, linear momentum, and angular momentum. Then, we use these to calculate the velocity and auxiliary variables such as inertia and omega (angular velocity). Further, we set one function to sum up the force and torque from other function or collision detection. And the friction part is implemented by comparing two points: one is fix point on the ground, and the other is the point on the surface. We use this relative velocity to determine the direction of friction force.
2.2 collision detection
∙ Approach overview
In order to achieve reasonable contact collision, we use the simplest method to determine whether two rigid bodies are closed enough. Then, we calculate the reaction between multiple objects.
∙ Implementation
First, we use more than one time iteration to derive the minimum tolerance of the distance of two colliding objects. After that, we use conservation of momentum and geometric principle to determine the later value of momentum. We also implement a circle constraint and point constraint to simulate the effect of ball’s swirling around the hole.
2.3 Visualization
∙ Introduction
We use OpenGL for rendering the whole billiard game and fltk for dealing with windows and user input.
∙ Implementation
The whole billiard game model is created by gl and glu calls. Each object consist these attributes: position, scale, rotate, and texture. Using display list, we manage the drawing functions of models and reduce the redundancy of drawing codes. The texture part is implemented by loading image file from outside .bmp files and mapping to the right coordinate on the objects.
3.0 Results
As you can see in the graphs, we create a billiard game with ten plus one balls and a billiard table. User can adjust the direction and scalar of the input force to the start ball. Once starting to play the game with keyboard, every ball interacts with the table boundaries and other balls.
Here is the link for download
our billiard game BilliardGame.rar
User input:
1. Shoot direction: number pad 4, 6, 2, 8: left right down up
2. Number pad : <Ctrl>+2, 8 change hit point to the Q-ball “+” increase power
3. Direction pad: ← ↑ → ↓ change
Q-ball position, if combine with
<Ctrl>+ ← ↑ → ↓, increase the moving step.
4. q: hit. r: reset Q-ball. <Shift>+ r : reset Q-ball and camera.
c:
switch the camera mode: free or follow Q-ball
4.0 References
An Introduction to Physically Based Modeling:
Rigid Body Simulation I—Unconstrained Rigid Body Dynamics
Rigid body simulation II – non penetration Constraints