CPSC 427 Video Game Programming Physics-Based Simulation

CPSC 427 Video Game Programming

Animation

• Movement governed by forces • Simple – Independent particles • Complex – Correct collisions, stacking, sliding 3D rigid bodies • Many many simulators! – PhysX (Unity, Unreal), Bullet, Open Dynamics Engine, MuJoCo, Havok, Box2D, Chipmunk, OpenSim, RBDL, Simulink (MATLAB), ADAMS, SD/FAST, DART etc…

• Particle systems – Fire, water, smoke, pebbles • Rigid-body simulation – Blocks, robots, humans • Continuum systems – Deformable solids – Fluids, cloth, hair • Group movement – Flocks, crowds © Alla Sheffer/M. van de Panne

Simulation Basics

Simulation loop… 1. Equations of Motion  sum forces & torques  solve for accelerations: 𝑭 𝒎𝒂 2. Numerical integration  update positions, velocities 3. Collision detection 4. Collision resolution

• Motion of one particle • Motion of many particles

𝑣 Second-order ODE 𝑥 𝜕 𝑥 𝑣 𝐹 /𝑚 𝐹⃑ 𝑚 𝜕𝑡 𝜕 𝑥 𝑣 𝑣 𝜕𝑡 𝐹/𝑚 First-order ODE ⋮ ⋮ 𝜕 𝑣⃑ 𝑥 𝑣 𝑥⃑ ⃑ 𝑣 𝜕𝑡 𝑣⃑ 𝛴𝐹/𝑚 𝐹/𝑚

Basic Particle Simulation (first try)

Forces only 𝒇 𝒎𝒂

𝒅𝒕 𝒕𝒊𝟏 𝒕𝒊

𝒗𝒊𝟏 𝒗 𝒕𝒊 𝒇𝒕𝒊/𝒎𝒅𝒕 𝒑𝒊𝟏 𝒑 𝒕𝒊 𝒗𝒕𝒊𝟏𝒅𝒕

Forces only 𝑭 𝒎𝒂

𝒅𝒕 𝒕𝒊𝟏 𝒕𝒊

𝒗𝒊𝟏 𝒗 𝒕𝒊 𝒇𝒕𝒊/𝒎𝒅𝒕 𝒑𝒊𝟏 𝒑 𝒕𝒊 𝒗𝒕𝒊𝟏𝒅𝒕

Basic Particle Forces

• Gravity 0 𝐹 𝑚𝑔

• Viscous damping 𝐹 𝑏𝑣

• Spring & dampers 𝐹𝑘𝑥 𝑏𝑣

• Behavior forces: • Fluids flocking birds, schooling fish, etc. [“Curl Noise for Procedural Fluid Flow” [“Boids”, Craig Reynolds, SIGGRAPH 1987] R. Bridson, J. Hourihan, M. Nordenstam, Proc. SIGGRAPH 2007]

Basic Particle Simulation: Small Problem…

Forces only 𝑭 𝒎𝒂

𝒅𝒕 𝒕𝒊𝟏 𝒕𝒊

𝒗𝒊𝟏 𝒗 𝒕𝒊 𝒇𝒕𝒊/𝒎𝒅𝒕 𝒑𝒊𝟏 𝒑 𝒕𝒊 𝒗𝒕𝒊𝟏𝒅𝒕

Equations of motion describe state (equilibrium)

Use: get values at time 𝒕𝒊𝟏from values at time 𝒕𝒊

• Motion of one particle • Motion of many particles

Ordinary Differential Equations

𝜕 𝑋⃑𝑡 𝑓𝑋⃑ 𝑡,𝑡 𝜕𝑡

Given that 𝑋⃑ 𝑋⃑ 𝑡

Compute 𝑋⃑ 𝑡 for t𝑡 ∆𝑋⃑𝑡 𝑓𝑋⃑ 𝑡,𝑡∆𝑡 • Simulation: – path through state-space – driven by vector field

𝜕 𝑋⃑𝑡 𝑓𝑋⃑ 𝑡,𝑡 𝜕𝑡

Given that 𝑋⃑ 𝑋⃑ 𝑡 𝒇𝑿 𝒕,𝒕 Compute 𝑋⃑ 𝑡 for tt

t t ∆𝑡 𝑿 𝒕∆𝒕 ∆𝒕 𝒇𝑿 𝒕,𝒕 ∆𝑋⃑ 𝑡∆𝑡 𝑓𝑋⃑ 𝑡,𝑡 𝑿 𝒕 𝑋⃑ 𝑋⃑ + ∆𝑡 𝑓𝑋⃑,t

ODE Numerical Integration: Explicit (Forward) Euler

𝜕 𝑋⃑𝑡 𝑓𝑋⃑ 𝑡,𝑡 𝜕𝑡

Given that 𝑋⃑ 𝑋⃑ 𝑡 𝒇𝑿 𝒕,𝒕 Compute 𝑋⃑ t for t t

∆𝑡 t t 𝑿 𝒕∆𝒕 ∆𝒕 𝒇𝑿 𝒕,𝒕 ∆𝑋⃑ t ∆𝑡 𝑓𝑋⃑ t ,t 𝑿 𝒕 𝑋⃑ 𝑋⃑ + ∆𝑡 𝑓𝑋⃑,t

• Solution spirals out – Even with small time steps – Although smaller time steps are still better

Explicit Euler Problems

• Can lead to instabilities

1. ½ Euler step 2. evaluate f at 𝑿 m 𝒎 𝒇𝑿 𝒕,𝒕 3. full step using fm

𝒇𝒎

𝑿𝒎 𝑿 𝒕 ½ ∆𝑿𝒕

½ ∆𝒕 𝒇 𝑿 𝒕,𝒕

𝑿 𝒕∆𝒕𝒎 𝒇 𝑿 𝒕

Trapezoid Method

1. full Euler step get 𝑿𝒂

2. evaluate ft at 𝑿𝒂 𝒇𝑿 𝒕,𝒕

3. full step using ft get 𝑿𝒃 𝒇𝒕 4. average 𝑿𝒂 and 𝑿𝒃 𝑿𝒂 𝑿∆𝑿𝒕

∆𝒕 𝒇𝑿 𝒕,𝒕 ½𝑿𝒂 ½𝑿𝒃

𝑿 𝒕 𝑿𝒃 𝑿 𝒕∆𝒕𝒇𝒕 © Alla Sheffer/M. van de Panne

• Not exactly the same – But same order of accuracy

𝑻𝒓𝒂𝒑𝒆𝒛𝒐𝒊𝒅 𝒎𝒆𝒕𝒉𝒐𝒅

𝑴𝒊𝒅𝒐𝒊𝒏𝒕 𝒎𝒆𝒕𝒉𝒐𝒅 𝑿 𝒕

Explicit Euler: Code

𝒇𝒎

𝑿𝒎 𝑿 𝒕 ½ ∆𝑿𝒕

½ ∆𝒕 𝒇 𝑿 𝒕,𝒕

𝑿 𝒕∆𝒕𝒎 𝒇 𝑿 𝒕

Implicit (Backward) Euler:

• Use the derivative at the destination

Forward Euler

𝒙𝒏𝟏 𝒙𝒏 𝒉 𝒏𝒗 𝒌 𝒙 𝒗 𝒗 𝒉 𝒏 𝒏𝟏 𝒏 𝒎 Backward Euler

𝒙𝒏𝟏 𝒙𝒏 𝒉 𝒏𝟏𝒗 • Problem is we don’t know the 𝒌 𝒙 𝒗 𝒗 𝒉 𝒏𝟏 destination yet 𝒏𝟏 𝒏 𝒎 © Alla Sheffer/M. van de Panne

• Use the derivative at the destination

Forward Euler

𝒙𝒏𝟏 𝒙𝒏 𝒉 𝒏𝒗 𝒌 𝒙 𝒗 𝒗 𝒉 𝒏 𝒏𝟏 𝒏 𝒎 Backward Euler

𝒙𝒏𝟏 𝒙𝒏 𝒉 𝒏𝟏𝒗 • Problem is we don’t know the 𝒌 𝒙 𝒗 𝒗 𝒉 𝒏𝟏 destination yet 𝒏𝟏 𝒏 𝒎 © Alla Sheffer/M. van de Panne

Simulation Basics

Simulation loop… 1. Equations of Motion 2. Numerical integration 3. Collision detection 4. Collision resolution

• Collision detection – Broad phase: AABBs, bounding spheres – Narrow phase: detailed checks • Collision response – Collision impulses – Constraint forces: resting, sliding, hinges, ….

Particle-Plane Collisions

• Particle-plane frictionless impulse response – Invert & scale perpendicular velocity component

𝒗 𝝐𝒗 𝒏 𝒗∥ 𝒗∥

𝒗 𝒗

• More formally… – Apply an impulse of magnitude j Inversely proportional to mass of particle – In direction of normal 𝒏 𝒋 𝟏𝝐 𝒎 ⃑ 𝒋 𝒏

⃑ 𝒗 𝒗 𝒗 𝒗 𝒎

Particle-Particle Collisions

• Particle-particle frictionless elastic impulse response

𝒗 𝒗𝟏 𝟏 𝒗 𝒎 𝒎𝟏 𝟐 𝟏 𝒗𝟐 𝒎 𝒎𝟐 𝟐 Before After

• Momentum is preserved 𝒎𝟏𝒗𝟏 𝒎𝟐𝒗𝟐 𝒎𝟏𝒗𝟏 𝒎𝟐𝒗𝟐 • Kinetic energy is preserved 𝟐 𝟐 𝟐 𝟐 ½ 𝒎𝟏𝒗𝟏 ½ 𝒎𝟐𝒗𝟐 ½ 𝒎𝟏𝒗𝟏 ½ 𝒎𝟐𝒗𝟐 © Alla Sheffer/M. van de Panne

• What we know… – Particle centers 𝒗𝟐 – Initial velocities 𝒑𝟐: 𝒙𝟐,𝒚𝟐 – Particle Masses

• What we can calculate… 𝒎𝟐 𝒑 : 𝒙 ,𝒚 – Contact normal 𝟏 𝟏 𝟏 𝒗𝟏 – Contact tangent

𝒎𝟏

Particle-Particle Collisions (radius >0)

• Impulse direction 𝒗𝟐

reflected across tangent 𝒗𝟐

• Impulse magnitude 𝒑𝟐: 𝒙𝟐,𝒚𝟐 proportional to mass of other 𝒎𝟐 particle 𝒑 : 𝒙 ,𝒚 𝟏 𝟏 𝟏 𝒗𝟏

• More formally…

𝟐𝒎𝟐 𝒗𝟏 𝒗𝟐 · 𝒑𝟏 𝒑𝟐 𝒗𝟏 𝒗𝟏 𝟐 𝒑𝟏 𝒑𝟐 𝒎𝟏 𝒎𝟐 𝒑𝟏 𝒑𝟐

𝟐𝒎𝟏 𝒗𝟐 𝒗𝟏 · 𝒑𝟐 𝒑𝟏 𝒗𝟐 𝒗𝟐 𝟐 𝒑𝟐 𝒑𝟏 𝒎𝟏 𝒎𝟐 𝒑𝟐 𝒑𝟏

Rigid Body Dynamics

• From particles to rigid bodies…

Particle Rigid body

𝒙 𝒑𝒐𝒔𝒊𝒕𝒊𝒐𝒏 𝒙 𝒑𝒐𝒔𝒊𝒕𝒊𝒐𝒏 𝒔𝒕𝒂𝒕𝒆 𝒗 𝒗𝒆𝒍𝒐𝒄𝒊𝒕𝒚 𝒗 𝒗𝒆𝒍𝒐𝒄𝒊𝒕𝒚 𝒔𝒕𝒂𝒕𝒆 𝒒, 𝑹 𝒓𝒐𝒕𝒂𝒕𝒊𝒐𝒏 𝒎𝒂𝒕𝒓𝒊𝒙𝟑𝒙𝟑 ℝ𝟒 in 2D 𝒘 𝒂𝒏𝒈𝒖𝒍𝒂𝒓 𝒗𝒆𝒍𝒐𝒄𝒊𝒕𝒚 ℝ𝟔 in 3D ℝ𝟏𝟐 in 3D © Alla Sheffer/M. van de Panne

• From particles to rigid bodies…

Newton’s equations of motion Newton-Euler equations of motion 𝚺𝑭 𝒎𝒂 𝒎 𝒂𝒙 𝒎 𝒂𝒚 𝚺𝑭 𝒂 𝒎 𝒂𝒛 𝒎 𝒙 𝒎 𝒂𝒚 𝚺𝑭 𝒘𝒙 𝒎 𝒂𝒛 𝚰 𝒘𝒚 𝒘𝒛 𝑴𝒂 𝚺𝑭 Inertia tensor 𝚺𝝉 𝒘𝚰𝒘

Resources

• Non-convex rigid bodies with stacking 3D collision processing and stacking http://www.cs.ubc.ca/~rbridson/docs/rigid_bodies.pdf

• Physically-based Modeling, course notes, SIGGRAPH 2001, Baraff & Witkin http://www.pixar.com/companyinfo/research/pbm2001/

• Doug James CS 5643 course notes http://www.cs.cornell.edu/courses/cs5643/2015sp/

• Rigid Body Dynamics, Chris Hecker http://chrishecker.com/Rigid_Body_Dynamics

• Video game physics tutorial https://www.toptal.com/game/video-game-physics-part-i-an-introduction-to-rigid-body-dynamics

• Box2D javascript live demos http://heikobehrens.net/misc/box2d.js/examples/

• Rigid body collisions javascript demo https://www.myphysicslab.com/engine2D/collision-en.html

• Rigid Body Collision Reponse, Michael Manzke, course slides https://www.scss.tcd.ie/Michael.Manzke/CS7057/cs7057-1516-09-CollisionResponse-mm.pdf

• Interactive simulation of rigid body dynamics in computer graphics, CGF 2014 http://onlinelibrary.wiley.com/doi/10.1111/cgf.12272/abstract

• A Mathematical Introduction to Robotic Manipulation (textbook) http://www.cds.caltech.edu/~murray/books/MLS/pdf/mls94-complete.pdf