1. Advanced Computer Graphics Rigid Body Simulation
Spring 2002
Professor Brogan

2. Upcoming Assignments Who wants a midterm instead of an assignment?
Final will be take home
Cloth/water/parallel particle sim presentations
Volunteers?
Papers selected by Thursday

3. Physical Simulation References Text book (4.3 and Appendix B)
Physics for Game Developers (Bourg)
Chris Hecker Game Developer articles

4. Equations of Motion
The physics-based equations that define how objects move
Gravity
Turbulence
Contact forces with objects
Friction
Joint constraints

5. Equations of Motion Equations of Motion Current State
Position and velocity
Forces
Integrate
Integrate
Velocities
Accelerations
Integrate

6. Equations of Motion Linear motions: Example:
Constant acceleration of 5 m/s2

7. Linear Momentum Mass times velocity = linear momentum, p
Newton’s Second Law
Ceasing to identify vectors…

8. Rigid Bodies Imagine a rigid body as a set of point masses
Total momentum, pT, is sum of momentums of all points:
Center of Mass (CM) is a single point. Vector to CM is linear combination of vectors to all points in rigid body weighted by their masses, divided by total mass of body

9. Total Momentum Rewrite total momentum in terms of CM
Total linear momentum equals total mass times the velocity of the center of mass
(For continuous rigid bodies, all summations turn into integrals over the body, but CM still exists)
We can treat all bodies as single point mass and velocity

10. Total Force Total force is derivative of the total momemtum
Again, CM simplifies total force equation of a rigid body
We can represent all forces acting on a body as if their vector sum were acting on a point at the center of mass with the mass of the entire body

11. Intermediate Results
Divide a force by M to find acceleration of the center of mass
Integrate acceleration over time to get the velocity and position of body
Note we’ve ignored where the forces are applied to the body
In linear momentum, we don’t keep track of the angular terms and all forces are applied to the CM.

12. Ordinary Differential Equations
A DifEq is an equation with
Derivatives of the dependent variable
Dependent variable
Independent variable
Ex:
v’s derivative is a function of its current value
Ordinary refers to ordinary derivatives
As opposed to partial derivatives

13. Integrating ODEs Analytically solving ODEs is complicated
Numerically integrating ODEs is much easier (in general)
Euler’s Method
Runge-Kutta

14. Euler’s Method
Based on calculus definition of first derivative = slope

15. Euler’s Method
Use derivative at time n to integrate h units forward

16. Euler Errors Depending on ‘time step’ h, errors will accumulate
SIGGRAPH Course Notes

17. Accumulating Errors
SIGGRAPH Course Notes

18. Recap

19. Angular Effects Let’s remain in 2-D plane for now
In addition to kinematic variables
x, y positions
Add another kinematic variable
W angle
CCW rotation of object axes relative to world axes

20. Angular Velocity w is the angular velocity
a is the angular acceleration

21. Computing Velocities How do we combine linear and angular quantities?
Consider velocity of a point, B, of a rigid body rotating about its CM
r_perp is perpendicular to r vector from O to B
Velocity is w-scaled perpendicular vector from origin to point on body

22. More Details Point B travels W radians Point B travels C units
Radius of circle is r
C= Wr
By definition of radians, where circumference = 2pr
B’s speed (magnitude of velocity vector)
Differentiate C= Wr w.r.t. time

23. More Details
The direction of velocity is tangent to circle == perpendicular to radius
Therefore, linear velocity is angular velocity multiplied by tangent vector

24. Chasles’ Theorem Any movement is decomposed into:
Movement of a single point on body
Rotation of body about that point
Linear and Angular Components

25. Angular Momentum
The angular momentum of point B about point A (we always measure angular terms about some point)
It’s a measure of how much of point B’s linear momentum is rotating around A

26. Angular Momentum
Check: If linear momentum, pB, is perpendicular to r_perp, then dot product of two will cause angular momentum will be zero

27. Torque Derivative of Angular Momentum
Remember force was derivative of linear momentum
This measures how much of a force applied at point B is used to rotate about point A, the torque

28. Total Angular Momentum
Total angular momentum about point A is denoted LAT
But computation can be expensive to sample all points
Sampling of a surface would require surface integration

29. Moment of Inertia Remember
An alternate way of representing the velocity of a point in terms of angular velocity
If A is like the origin i is like B, then substitute

30. Moment of Inertia, IA
The sum of squared distances from point A to each other point in the body, and each squared distance is scaled by the mass of each point
This term characterizes how hard it is to rotate something
Ipencil_center will be much less than Ipencil_tip

31. Total Torque Differentiate total angular momentum to get total torque
This relates total torque and the body’s angular acceleration through the scalar moment of inertia

32. Planar Dynamics Calculate COM and MOI of rigid body
Set initial position and linear/angular velocities
Figure out all forces and their points of application
Sum all forces and divide by mass to find COM’s linear acceleration
For each force, compute perp-dot-product from COM to point of force application and add value into total torque of COM
Divide total torque by the MOI at the COM to find angular acceleration
Numerically integrate linear/angular accelerations to update the position/orientation and linear/angular velocities
Draw body in new position and repeat

33. Upcoming Topics Collisions
3-dimensional rigid bodies (inertia tensors)
Forces (centripetal, centrifugal, viscosity, friction, contact)
Constrained dynamics (linked bodies)