Advanced Computer Graphics Rigid Body Simulation Spring 2002 Professor Brogan

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

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

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

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

Equations of Motion Linear motions: Example: –Constant acceleration of 5 m/s 2

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

Rigid Bodies Imagine a rigid body as a set of point masses Total momentum, p T, 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

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

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

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.

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

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

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

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

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

Accumulating Errors SIGGRAPH Course Notes

Recap

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

Angular Velocity  is the angular velocity  is the angular acceleration

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  -scaled perpendicular vector from origin to point on body

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

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

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

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

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

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

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

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

Moment of Inertia, I A 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 –I pencil_center will be much less than I pencil_tip

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

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

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