aselab PHYSICS ENGINE Rigid body simulator introduction Continuum

東 大長谷川(晶)研 aselab PHYSICS ENGINE Rigid body simulator introduction Continuum simulation The Documents http: //haselab. net/class/vr user：basic　passwd：prog

Introduction for rigid body physics engine

aselab 東 大長谷川(晶)研 What is physics engine ? n Usual modeling of dynamics and simulation n Manual formulation q q Find this equation with mysterious algorithm in human brain. Lagrangian also requires some human intelligence to find formulation of energy simulate this equation by ODE Solver

aselab 東 大長谷川(晶)研 What is physics engine ? Physics engine n Forward dynamics is automatically computed from the configuration of the scene with formal algorithm. n fc q n Other than constraint force fc , Newton and Euler’s equation requires external force and it’s application point only. So, physics engine calculate constraint force fc with formal algorithm. ⇒ Human intelligence is no longer required.

aselab 東 大長谷川(晶)研 Rigid body motion v: velocity w: angular velocity m: mass I: inertial tensor f: external force Equation of motion Velocity and angular velocity are constant in case 5

aselab 東 大長谷川(晶)研 Rigid body motion simulation n Force f

aselab 東 大長谷川(晶)研 Procedure of physics simulation 1. Find contact points 2. Calculate the contact forces 3. Update the positions and velocities Talk about this in detail next lesson

aselab 東 大長谷川(晶)研 Taxonomy of physics simulator n Computation Methods for constraint forces are not limited to Spring and Damper. Multi Body Dynamics simulator Analytical method Impulse method Full coordinate method Hahn 88 Mirtich 96 Jean Jacques Moreau method LCP based formularization Acceleration Stewart 96 ODE n Reduced coordinate method Moore & Williams 89 Springhead 1 Velocity base Baraff 89 penalty method Armstrong 79 Featherstone 83 Springhead 2 Open. Tissue Velocity based LCP is widely used. n Havok Novodex ODE … reference： Kenny Erleben: Multibody Dynamics Animation Lecture 12 Department of Computer Science Copenhagen University 8

aselab 東 大長谷川(晶)研 Rigid body motion equation n Two objects＋Restraint＋External Force Constraint mj Ij vj wj ri Inertia M Velocity and Angular velocity rj mi Ii vi wi n Many rigid bodies＋constraints ＋ external forces u force fc Other external force, gravity fe

aselab 東 大長谷川(晶)研 Constraint Examples n Joint(Hinge) (Contacts are mentioned later) : relative velocity/angular velocity at the joint point w l j : constraint forces and torques on the joint point w 1 , l 1 i e 1 w 4 , l 4 w 6 , l 6 w 3 , l 3 e 2 w 2, , l 2 w 5 , l 5 Constraint coordinate system n n velocity Only rotate the hinge axis e 3 : Rotate hinge axis e 3 freely = torque not work： force velocity force 10

aselab 東 大長谷川(晶)研 Embed constraints into equation of motion n Express equation of motion on the constraint coordinate. vi , wi: velocity or angular velocity of object i in world coordinate. → use w, l w: relative velocity and angular velocity of j from i in constraint coordinate vj , wj rj w ri Jrow 1 vi , wi u Matrix expression of cross product Jrow 4 u express w 2 w 3, w 5 w 6, with the same way 11

aselab 東 大長谷川(晶)研 Embed constraints into equation of motion vi , wi: velocity, angular velocity of object i vj , wj: velocity, angular velocity of object j vj , wj rj w ri w velocity/angular u velocity/angular vi , wi velocity in constraint of rigid body coordinate fj , tj: constraint forces given from i to j in world coordinate lj : constraint forces given from i to j in constraint coordinate fj , tj rj l ri fi , ti fc force/torque in world coordinate l: force/torque in constraint coordinate 12

aselab 東 大長谷川(晶)研 Embed constraints n Transform equation of motion to consider constraint easily n Ally with constraint condition (hinge’s case) Constraint： Substitute to equation of motion： Solve the simultaneous equations 13

aselab 東 大長谷川(晶)研 Simulation Procedure n Calculate constraint force l n Update rigid body velocity n Update rigid body position posture of all rigid bodies position orientation quaternion A matrix transforming velocity into Quaternion differential 14

aselab 東 大長谷川(晶)研 Contact Constraint n Contact relative velocity at joint point contact force on contact point n n Do not penetrate each other ＝ velocity l is not zero when w is zero or l is zero when w is not zero: force Static friction or kinetic friction： velocity force n Torque is 0： velocity force 15

aselab 東 大長谷川(晶)研 Embed Constraint (for contact) n Transform motion equation to consider restraint easily n Ally with constraint condition velocity force At first, solve this case and find li. If li does not satisfy wi = 0 then fix li to a constant and solve wi There are two variables in an equation at a glance, but actually solve one variable with consider the other variable as a constant. 16

aselab 東 大長谷川(晶)研 Summary n Velocity based LCP is widely used. n Formulate constraints into velocity based LCP and find constraint force. Then do the stepping of the simulation Transform equation of motion of solids with Jacobian J → Equation of motion by w and l. n Formulate constraints by equations of w and l and substitute it into equation of motion. → The equations come to LCP n Solve LCP and find l update velocity and position. n n Constraints Joints are lines, contacts are polylines. n It looks there are two variables. But one of the variables comes to constant. n 17

aselab 東 大長谷川(晶)研 Computation n. Simultaneous n. Approximate equations solution for LCP n. Error-correction method 18

aselab 東 大長谷川(晶)研 How to Solve Simultaneous Equation n Restraint included motion equation two variable in one equation +complementary condition of wi and l i (one is a variable and the other is a constant ) LCP : Linear Complementary Problem n Solve LCP Pivoting method　・・・strict slow speed n Iterative method　・・・high speed, precision depends on iteration number of times n n Matrix-splitting method Jacobi method n Gauss–Seidel method, SOR method n n Newton's method 19

aselab 東 大長谷川(晶)研 Gauss–Seidel method / SOR method n Matrix-splitting method n A method to solve large simultaneous equations approximately with small computation amount. If recurrence relation is converged, So This recurrence relation converge if A is positive symmetry matrix. Long narrow objects or objects with small inertia converge slow. Choose D as a simple matrix such as diagonal matrix n Jacobi method choose D as diagonal entries of A and choose F as the rest. Gauss-Seidel method is improved version of Jacobi method. n 20

aselab 東 大長谷川(晶)研 Gauss–Seidel method / SOR method n Gauss–Seidel method Jacobi method：update l at once Gauss–Seidel method ：update l one row at one time renewed under renewing not-renewed n SOR method n Just accelerate Gauss–Seidel method slightly Gauss–Seidel method 1. 6 times acceleration n Converge quickly 21

aselab 東 大長谷川(晶)研 LCP and Gauss–Seidel method n Gauss–Seidel method renew l one row at one time renewed under renewing not-renewed n Calculate Contact force perfectly not invade each other＝ l is not zero when ω is zero or l is zero when ω is not zero: n　Do if l 1 is renewed to a negative number, then let l 1=0 n　Static friction or kinetic friction： Use renewed l 1 to check “-ml 1 <l 2< ml 1”, if protrude, clip with l 1=+-m l 1 22

aselab 東 大長谷川(晶)研 Simulation examples 23

Continuum simulation

aselab 東 大長谷川(晶)研 Continuum simulation n Mass point v n Coil voltage t V t Continuum Simulation is needed not only in time but also in space dimension. n wave n n solid Discretize Hooke‘s law and equation of motion n fluid t x

aselab 東 大長谷川(晶)研 Continuum simulation n Continuum simulation continuous →deal with limited representative points not infinity n many methods for taking representative points n n Grid points in Euler coordinate system are representative points Common for fluid and wave surface simulation n Grid do not move n Mass come from surroundings and go out n Grid points in Lagrange coordinate system are representative points n Finite element method n Grids move with mass n Grids collapse or turn over under large deformation n n Particle method： representative points are particles Express mass with freely moving particles n Large deformation is ok because of no assumption at beginnings n Collision detection is important n

aselab 東 大長谷川(晶)研 Finite Element Method n. Soft objects（meat, jelly, human body）

aselab 東 大長谷川(晶)研 Soft objects ncloth n. Mass point＋ Spring-damper model n. Omit parts of stress

aselab 東 大長谷川(晶)研 Fluid n. Liquid n. Gas

aselab 東 大長谷川(晶)研 Model what? n Wave Particles

aselab 東 大長谷川(晶)研 Model what? n Vortex field of fluid

aselab 東 大長谷川(晶)研 Speed and stability n Rigid body stability is more important than speed n The number of rigid bodies is almost enough n Any simulation oscillates n Guarantee control stability Havok FX example PD control oscillation example Wrotek et al. 2006 32

aselab 東 大長谷川(晶)研 Speed and stability n The speed of fluid is not fast enough n 2 D wave front → sufficient real-time 3 D mesh → more difficult n 3 D particles → can not deal with many n Combination is needed possibly n n A Fluid Resistance Map Method for Real-time Haptic Interaction with Fluids [Y. Dobashi et al. 　2006] 3 D-fluid simulation 圧力場・流速 (FRM : Fluid Resistance Map) Prior calculation Paddle speed Water speed R Force feedback n Real-time wave simulation Real-time Efficient Simulation of Large Bodies of Water by Coupling Two- and Three-Dimensional Techniques [G. Irving et al. 2006] 33

aselab 東 大長谷川(晶)研 High speed n Soft objects which can be simulated are numerous n Record and reproduce the deformation caused by impulse force n Modal analysis based object deformation