Advanced Computer Graphics Rigid Body Simulation

1. Advanced Computer GraphicsRigid 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 • http://www.d6.com/users/checker/dynamics.htm

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 ofMotion 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 asingle point. Vector to CMis linear combination ofvectors to all points in rigidbody 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 objectaxes 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, linearvelocity is angularvelocity 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 origini 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 totalangular momentumto 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)