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Download Dynamics 101 - Essential Math for Games Programmers
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Dynamics 101 Marq Singer Red Storm Entertainment [email protected] 2 Talk Summary Going to talk about: A brief history of motion theory Newtonian motion for linear and rotational dynamics Handling this in the computer 3 Physically Based-Motion Want game objects to move consistent with world Match our real-world experience But this is a game, so… Can’t be too expensive (no atomic-level interactions) 4 History I: Aristotle Observed: From this, deduced: Push an object, stop, it stops Rock falls faster than feather Objects want to stop Motion is in a line Motion only occurs with action Heavier object falls faster Note: was not actually beggar for a bottle 5 History I: Aristotle Motion as changing position 6 History I: Aristotle Called kinematics Games: move controller, stop on a dime, move again Not realistic 7 History II: Galileo Observed: Object in motion slows down Cannonballs fall equally Theorized: Slows due to unseen force: friction Object in motion stays in motion Object at rest stays at rest Called inertia Also: force changes velocity, not position Oh, and mass has no effect on velocity 8 History II: Galileo Force as changing velocity Velocity changes position Called dynamics 9 History III: Newton Observed: Planet orbit like continuous falling Theorized: Planet moves via gravity Planets and small objects linked Force related to velocity by mass Calculus helps formulate it all 10 History III: Newton Sum of forces sets acceleration Acceleration changes velocity Velocity changes position g 11 History III: Newton Games: Move controller, add force, then drift 12 History III: Newton As mentioned, devised calculus Differential calculus: rates of change Integral calculus: (concurrent with Leibniz) areas and volumes antiderivatives Did not invent the Fig Newton™ 13 Differential Calculus Review Have position function x(t) Derivative x'(t) describes how x changes as t changes (also written dx/dt, or x ) x'(t) gives tangent vector at time t x(ti) y y(t) x'(ti) t 14 Differential Calculus Review Our function is position:x(t ) Derivative is velocity: dx v(t ) x(t ) x dt Derivative of velocity is acceleration dv v dt d 2x x(t ) 2 x dt a(t ) v(t ) 15 Newtonian Dynamics Summary All objects affected by forces Gravity Ground (pushing up) Other objects pushing against it Force determines acceleration (F = ma) dv a) Acceleration changes velocity ( dx dt Velocity changes position ( v ) dt 16 Dynamics on Computer Break into two parts Linear dynamics (position) Rotational dynamics (orientation) Simpler to start with position 17 Linear Dynamics Simulating a single object with: Last frame position xi Last frame velocity vi Mass m Sum of forces F Want to know Current frame position xi+1 Current frame velocity vi+1 18 Linear Dynamics Could use Newton’s equations a F /m x i1 x i v i t 1/2at 2 v i1 v i at Problem: assumes F constant across frame Not always true: E.g. spring force: Fspring = –kx E.g. drag force: Fdrag = –mv 19 Linear Dynamics Need numeric solution Take stepwise approximation of function 20 Linear Dynamics Basic idea: derivative (velocity) is going in the right direction Step a little way in that direction (scaled by frame time h) Do same with velocity/acceleration Called Euler’s method 21 Linear Dynamics Euler’s method x i1 x i hv i a F(x i ,v i ) /m v i1 v i ha 22 Linear Dynamics Another way: use linear momentum P mv P F Then x i1 x i hvi Pi1 Pi hF(x i ,v i ) v i1 Pi1 /m 23 Linear: Final Formulas Using Euler’s method with time step h F Fk (x i ,v i ) k F ai m x i1 x i hv i v i1 v i ha i 24 Rotational Dynamics Simulating a single object with: Last frame orientation Ri or qi Last frame angular velocity i Inertial tensor I Sum of torques Want to know Current frame orientation Ri+1 or qi+1 Current frame ang. velocity i+1 25 Rotational Dynamics Orientation Represented by Rotation matrix R Quaternion q Which depends on your needs Hint: quaternions are cheaper 26 Rotational Dynamics Angular velocity Represents change in rotation How fast object spinning 3-vector Direction is axis of rotation Length is amount of rotation (in radians) Ccw around axis (r.h. rule) 27 Rotational Dynamics Angular velocity Often need to know linear velocity at point Solution: cross product r v v r 28 Moments of Inertia Inertial tensor I is rotational equivalent of mass 3 x 3 matrix, not single scalar factor (unlike m) Many factors - rotation depends on shape Describe how object rotates around various axes Not always easy to compute Change as object changes orientation 29 Rotational Dynamics Computing I Can use values for closest box or cylinder Alternatively, can compute based on geometry Assume constant density, constant mass at each vertex Solid integral across shape See Mirtich,Eberly for more details Blow and Melax do it with sums of tetrahedra 30 Rotational Dynamics Torque Force equivalent Apply to offset from center of mass – creates rotation Add up torques just like forces 31 Rotational Dynamics Computing torque Cross product of vector r (from CoM to point where force is applied), and force vector F r F r F Applies torque ccw around vector (r.h. rule) 32 Rotational Dynamics Center of Mass Point on body where applying a force acts just like single particle “Balance point” of object Varies with density, shape of object Pull/push anywhere but CoM, get torque Generally falls out of inertial tensor calculation 33 Rotational Dynamics Have matrix R and vector How to compute Ri1 Ri hi ? Convert to give change in R Convert to symmetric skew matrix Multiply by orientation matrix Can use Euler's method after that 34 Computing New Orientation If have matrix R, then ˜ iRi Ri1 Ri h where 0 ˜ 3 2 3 0 1 2 1 0 35 Computing New Orientation If have quaternion q, then h q i 1 q i w i q i 2 where wi (0,i ) See Baraff or Eberly for derivation 36 Computing Angular Velocity Can’t easily integrate angular velocity from angular acceleration: I I I Can no longer “divide” by I and do Euler step 37 Computing Angular Momentum Easier way: use angular momentum L I L Then L i1 L i h (Ri , i ) i1 I L i1 1 i1 38 Using I in World Space Remember, I1L I computed in local space, must transform to world space If using rotation matrix R, use formula 1 i 1 i 0 I Li R I R Li T i If using quaternion, convert to matrix 39 Rotational Formulas rk Fk k ~ R i 1 R n hi R i L i 1 L i h I 1 i 1 1 i 1 0 R I R i 1 I i11L i 1 T i 1 40 Impulses Normally force acts over period of time F E.g., pushing a chair t 41 Impulses Even if constant over frame F sim assumes application over entire time t 42 Impulses But if instantaneous change in velocity? Discontinuity! F t Still force, just instantaneous Called impulse - good for collisions/constraints 43 Summary Basic Newtonian dynamics Linear simulation Position, velocity, force, momentum Force -> acceleration -> velocity -> position Rotational simulation Torque -> ang. mom. -> ang. vel. -> orientation 44 Questions? 45 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.
