1 / 26

CS 378: Computer Game Technology

CS 378: Computer Game Technology. Physics for Games Spring 2012. Game Physics – Basic Areas. Point Masses Particle simulation Collision response Rigid-Bodies Extensions to non-points Soft Body Dynamic Systems Articulated Systems and Constraints Collision Detection. Physics Engines.

elroy
Download Presentation

CS 378: Computer Game Technology

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. CS 378: Computer Game Technology Physics for Games Spring 2012

  2. Game Physics – Basic Areas • Point Masses • Particle simulation • Collision response • Rigid-Bodies • Extensions to non-points • Soft Body Dynamic Systems • Articulated Systems and Constraints • Collision Detection

  3. Physics Engines • API for collision detection • API for kinematics (motion but no forces) • API for dynamics • Examples • Box2d • Bullet • ODE (Open Dynamics Engine) • PhysX • Havok • Etc. University of Texas at Austin CS 378 – Game Technology Don Fussell 3

  4. Particle dynamics and particle systems • A particle system is a collection of point masses that obeys some physical laws (e.g, gravity, heat convection, spring behaviors, …). • Particle systems can be used to simulate all sorts of physical phenomena: University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell 4

  5. Particle in a flow field • We begin with a single particle with: • Position, • Velocity, • Suppose the velocity is actually dictated by some driving function g: y x g(x,t) University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell 5

  6. Vector fields • At any moment in time, the function g defines a vector field over x: • How does our particle move through the vector field? University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell 6

  7. Diff eqs and integral curves • The equation is actually a first order differential equation. • We can solve for x through time by starting at an initial point and stepping along the vector field: • This is called an initial value problem and the solution is called an integral curve. Start Here University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell 7

  8. Eulersmethod • One simple approach is to choose a time step, Dt, and take linear steps along the flow: • Writing as a time iteration: • This approach is called Euler’s method and looks like: • Properties: • Simplest numerical method • Bigger steps, bigger errors. Error ~ O(Dt2). • Need to take pretty small steps, so not very efficient. Better (more complicated) methods exist, e.g., “Runge-Kutta” and “implicit integration.”

  9. Particle in a force field • Now consider a particle in a force field f. • In this case, the particle has: • Mass, m • Acceleration, • The particle obeys Newton’s law: • The force field f can in general depend on the position and velocity of the particle as well as time. • Thus, with some rearrangement, we end up with: University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell 9

  10. Second order equations This equation: is a second order differential equation. Our solution method, though, worked on first order differential equations. We can rewrite this as: where we have added a new variable v to get a pair of coupled first order equations. University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell 10

  11. Phase space • Concatenate x and v to make a 6-vector: position in phase space. • Taking the time derivative: another 6-vector. • A vanilla 1st-order differential equation. University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell 11

  12. Differential equation solver Starting with: Applying Euler’s method: And making substitutions: Writing this as an iteration, we have: Again, performs poorly for large Dt. University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell 12

  13. Particle structure How do we represent a particle? Position in phase space position velocity force accumulator mass University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell 13

  14. Single particle solver interface getDim getState setState derivEval University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell 14

  15. n time particles Particle systems In general, we have a particle system consisting of n particles to be managed over time: University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell 15

  16. n time particles Particle system solver interface For n particles, the solver interface now looks like: get/setState getDim derivEval University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell 16

  17. Particle system diff. eq. solver We can solve the evolution of a particle system again using the Euler method: University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell 17

  18. Forces • Each particle can experience a force which sends it on its merry way. • Where do these forces come from? Some examples: • Constant (gravity) • Position/time dependent (force fields) • Velocity-dependent (drag) • Combinations (Damped springs) • How do we compute the net force on a particle? University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell 18

  19. Particle systems with forces • Force objects are black boxes that point to the particles they influence and add in their contributions. • We can now visualize the particle system with force objects: nf forces n time particles Fnf F1 F2 University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell 19

  20. Gravity and viscous drag The force due to gravity is simply: p->f += p->m * F->G Often, we want to slow things down with viscous drag: p->f -= F->k * p->v University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell 20

  21. r= rest length Damped spring Recall the equation for the force due to a spring: We can augment this with damping: The resulting force equations for a spring between two particles become: University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell 21

  22. derivEval Clear forces Loop over particles, zero force accumulators Calculate forces Sum all forces into accumulators Return derivatives Loop over particles, return v and f/m Clear force accumulators 1 Apply forces to particles 2 Fnf F1 F2 F3 3 Return derivativesto solver University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell 22

  23. P N Bouncing off the walls • Add-on for a particle simulator • For now, just simple point-plane collisions A plane is fully specified by any point P on the plane and its normal N. University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell 23

  24. x v N P Collision Detection How do you decide when you’ve crossed a plane? University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell 24

  25. v v N P Normal and tangential velocity To compute the collision response, we need to consider the normal and tangential components of a particle’s velocity. University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell 25

  26. Collision Response before after Without backtracking, the response may not be enough to bring a particle to the other side of a wall. In that case, detection should include a velocity check: University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell 26

More Related