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Rigid Body Dynamics Jim Van Verth ([email protected]) Rigid Body Dynamics • Simplest form of physical simulation • Gets you a good way towards making a more realistic looking game • Not that hard, either Essential Math for Games Rigid Body • Objects we simulate will not deform • Brick vs. clay • Fixed model: only change position and orientation Essential Math for Games Dynamics • Want to move objects through the game world in the most realistic manner possible • Applying velocity not enough – need ramp up, ramp down – acceleration • Same with orientation Essential Math for Games Calculus Review • Have function y(t) • Function y'(t) describes how y changes as t changes (also written dy/dt) • y'(t) gives slope at time t y y(t) y'(t) t Essential Math for Games Calculus Review • Our function is position: • Derivative is velocity: • Derivative of velocity is acceleration Essential Math for Games Basic Newtonian Physics • All objects affected by forces Gravity Ground (pushing up) Other objects pushing against it • Force determines acceleration (F = ma) • Acceleration changes velocity ( ) • Velocity changes position ( ) Essential Math for Games Basic Newtonian Physics • Assume acceleration constant, then • Similarly Essential Math for Games Basic Newtonian Physics • Key equations • Note: force is derivative of momentum P Remember for later – easier for angular Essential Math for Games Basic Newtonian Physics • General approach Compute all forces on object, add up Compute acceleration • (divide total force by mass) Compute new position based on old position, velocity, acceleration Compute new velocity based on old velocity, acceleration Essential Math for Games Newtonian Physics • Works fine if acceleration is constant • Not good if acceleration dependant on position or velocity – changes over time step • E.g. spring force: Fspring = –kx • E.g. drag force: Fdrag = –mv Essential Math for Games Analytic Solution • Can try and find an analytic solution I.e. a formula for x and v In case of simple drag: But not always a solution Or may want to try different simulation formulas Essential Math for Games Numeric Solution • Problem: Physical simulation with force dependant on position or velocity • Start at x(0) = x0, v(0) = v0 • Only know: • Basic solution: Euler’s method Essential Math for Games Euler’s Method • Idea: we have the derivative (x or v) • From calculus, know that • Or, for sufficiently small h: Essential Math for Games Euler’s Method • Can re-arrange as: • Gives us next function value in terms of current value and current derivative Essential Math for Games Final Formulas • Using Euler’s method with time step h Essential Math for Games What About Orientation? • Force (F) applies to center of mass* of object – creates translation • Torque () applies to offset from center of mass – creates rotation • Add up torques just like forces Essential Math for Games Force vs. Torque (cont’d) • To compute torque, take cross product of vector r (from CoM to point where force is applied), and force vector F • Applies torque ccw around vector r F Essential Math for Games Other Angular Equivalents • • • • • Force F vs. torque Momentum P vs. angular momentum L Velocity v vs. angular velocity Position x vs. orientation Mass m vs. moments of inertia J Essential Math for Games Why L? • Difficult to compute angular velocity from angular acceleration • Compute ang. momentum by integrating torque • Compute ang. velocity from momentum • Since then Essential Math for Games Moments of Inertia • Moments of inertia are 3 x 3 matrix, not single scalar factor (unlike m) • Many factors because rotation depends on shape and density • Describe how object rotates around various axes • Not easy to compute • Change as object changes orientation Essential Math for Games Computing J • Can use moments of inertia for closest box or cylinder • Can use sphere (one factor: 2mr2/5) • Or, can just consider rotations around one axis and fake(!) the rest • With the bottom two you end up with just one value… can simplify equations Essential Math for Games Computing J • Alternatively, can compute based on geometry • Assume constant density, constant mass at each vertex • Solid integral across shape • See Eberly for more details Also at www.geometrictools.com Essential Math for Games Using J in World Space • Remember, • J computed in local space, must transform to world space • If using rotation matrix , use formula • If using quaternion, convert to matrix Essential Math for Games Computing New Orientation • Have matrix and vector • How to integrate? • Convert to give change in Change to linear velocity at tips of basis vectors One for each basis gives 3x3 matrix Can use Euler's method then Essential Math for Games Computing New Orientation • Example: Essential Math for Games Computing New Orientation • r gives linear velocity at r • Could do this for each basis vector • Better way: Use symmetric skew matrix to compute cross products Multiply by orientation matrix Essential Math for Games Computing New Orientation • If have matrix , then where Essential Math for Games Computing New Orientation • If have quaternion q, then where • See Baraff or Eberly for derivation Essential Math for Games Computing New Orientation • We can represent wq as matrix multiplication where • Assumes q = (w, x, y, z) Essential Math for Games Angular Formulas Essential Math for Games Reducing Error • Keep time step as small as possible • Clamp accelerations, velocities to maximum values – avoid large forces • If velocity, acceleration very small, set to zero (avoids little shifts in position) • Damping acceleration based on velocity (i.e. friction) can help Essential Math for Games Improving Performance • If not moving, don’t simulate • Only do as much as you have to • If you can fake it, do so objects on ground, don’t bother with gravity only rotate around z, don’t bother with J simple drag instead of full friction model Essential Math for Games References • Burden, Richard L. and J. Douglas Faires, Numerical Analysis, PWS Publishing Company, Boston, MA, 1993. • Hecker, Chris, “Behind the Screen,” Game Developer, Miller Freeman, San Francisco, Dec. 1996-Jun. 1997. • Witken, Andrew, David Baraff, Michael Kass, SIGGRAPH Course Notes, Physically Based Modelling, SIGGRAPH 2002. • Eberly, David, Game Physics, Morgan Kaufmann, 2003. Essential Math for Games