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Computer Animation

by: Alayna Veum

Computer Animation CS 7496

Alayna Veum

GPA 3.81


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Class Notes
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This 0 page Class Notes was uploaded by Alayna Veum on Monday November 2, 2015. The Class Notes belongs to CS 7496 at Georgia Institute of Technology - Main Campus taught by Staff in Fall. Since its upload, it has received 13 views. For similar materials see /class/234132/cs-7496-georgia-institute-of-technology-main-campus in ComputerScienence at Georgia Institute of Technology - Main Campus.


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Date Created: 11/02/15
Munging motion capture data goal take captured motion and make it work on an arbitrary gure Without destroying the style of the motion Allow the animator control over the motion speed contact With environment style Problems cycli cationtransitions editing retargeting generalization extracting style Representation of data Fourier series wavelets spline with knot points Modi cation Technology smoothing of spline interpolation between two motions optimization with constraints Type of Modi cation style of motion cycli cation constraints on the motion transitions between motions Evaluation Criteria how many motions how good is the motion kinematic dynamic natural usable UI Editing Motion Warping Witkin and P0p0vic Siggraph 95 keyframe placing the ball on the racket at impact e t at t bt few keyframes possibly unrealistic motion Edltlng Motion Editing with Spacetime Constraints Gleicher 1997 Symposium on Interactive 3D Graphics FFame39dquot 39 FPame T0 7 9quot Frame39 7 iv Ffame 4 7 quot Fr ame 2 5 gt quot 39 FFame 37 a quot 7 7 7 7 7 7 7 7 QFrrr Frame 20 Frame 24 Frame 20 Frame 37 WW mid Editing Editing combine motion displacement techniques Witkin and Popovic with trajectory optimization spacetime constraints 1 interactively add constraints 2 minimize change in motion While maintaining constraints 3 allow user to interactively re ne minimize gX subject to fx c represent as mtX m0t dtx Retargeting accomodate size of character While retaining style 1998 Siggraph Gleicher Retargeting Editing A Hierarchical Approach to Interactive Motion Editing for Humanlike Figures Lee and Shin Siggraph 99 multi level bspline representation series of Bspline functions with different knot spacings that provide closer and closer approximations Keyframing based on hand animation techniques animator gt inbetweener gt inker Snow White 100week gt 35week gt 25week save time for the animator force explicit thought about timing vs straightahead animation simple interpolation intelligence judgment of inbetweener Keyframing ugly motion as Alan showed Fm a d2Xtdt2 gt C2 continuous xt position of ball Splines for interpolation Rotation Representation transformation matrices Rotx6 Roty6 multiply to get arbitrary rotations Rotz6 t AN RX t AN ltgtlt x y39 Z 1 1 0 0 0 0 cose isine 0 0 sine cost 0 0 0 0 1 c036 O sine 0 O 1 0 0 isine O 0056 O 7 O O O 17 icosB isine O 07 sine 0059 O 0 0 O 1 0 700 017 HNRR lwmax lr N39Vik Rotation Interpolation 3x3 rigid body rotation matrix has rows and columns that are orthonormal unit length and perpendicular simple interpolation won t preserve that property object won t rotate rigidly 1 0 0 0 x R0tX90 gtR0tX 90 3 c sg ism 3 y oint in the middle is 0 O 0 l 1 5 8 8 8 0 0 0 0 0 0 0 1 x y39 Zl l Rotation Euler Angles fixed order rotation about xyz 174289 if axes become aligned then representation breaks down gimbal lock xyz O900 incremental changes in either x or z Z have the same effect on the system A degree of freedom has been quotlostquot X Z Z l I xyz 9090 A90 Y Y X Y X will allow rotation by A about vertical Rotx0 Roty90 Rotz0 aXlS Interpolation of Euler Angles O900 gt 904590 is just a rotation by 45 in X but obvious interpolation would go through 45 675 45 which isn t the shortest path 2 Cloth Clothing 3 Collisions James F O39Bnen Page 1 Many types of cloth Very different properties Not a simple elastic surface Tendency to be very stiff Anisotropic Breen 95 1mg mmva 2 Resolution of Mesh P0ssible shapes Sm00thness Simulati0n time Breen 95 1mg mmva z f Resolution of Mesh Effects time step WWWW AtXl AtX2 AtX3 AtX4 James F O Brien Page 4 Seaming Clothes James F O39Bnen Page 5 inet egllgidiate Bounding Box Hierarchy Partition space or objects not the same Avoid expensive primitive tests Ol0gmn VS 0mn average not worst case A Top Level Tunneling JamesF O39Brien Page a Leaves in the Wind 1 mmva 7 gt l T 7K 7 Uniform Sink Source Vortex Add together to make interesting elds F F F F ocquot A V v F 061 v Wejchen amp Haumann 91 CC is coupling strength A area James F O Brien Page 8 Water Waves Splashes James F O Bn39en Page 1 0 Simulation Model Three part system Spray Surface Volume Interface between subsystems Fluid Volume Model Space divided into columns Flow through pipes between neighbors Driven by hydro static pressure Column is homogeneous impact at surface is immediately seen at bottom Fluid Volume Model g Gravity 1 Pipe length PhljpgP0Ei U 1 p Fluld denSIty pgh h 35 E h Column height if k ij kl pl External pressure Vi Volume of column i j AVlj 82 CLHz S aij kl 611 611 Pi Pressure in column i j aiHl Flow acceleration qud Flow rate ij to kl James F O Brien Page 1 3 Fluid Surface Model Mesh of control nodes Vertical position determined by volume in columns V zz0 dd x y d Node spacing z Vertical position of mesh James F O Brien Page 1 4 Spray Model Particle system Particles created when upward velocity exceeds a threshold Initial velocity is determined from volume and surface Total system volume remains constant James F O Brien Page 1 5 Object Interaction Computing precise contact forces 1s expens1ve Use approximation Compute bounding interval 2z so st fE T 3 Use heuristics to select value within range 0 m Apply force as external pressure James F O Brien Page 1 6 Kinematics the study of motion without regard to the forces that cause it 06 B A Forward A focB Inverse ocBflA draw graphics specify fewer degrees of freedom more intuitive control of dof contact with the environment calculate desired joint angles for control User Control of Kinematic Characters Ioint Space position all joints fine level of control Cartesian Space specify environmental interactions easily most dof computed automatically Forward Kinematics x Llcos 91 chos 91 92 y Llsm 91 Lzsm 9 1 9 2 HNltx r IOOO Why IK pick up an object or place feet on the ground hard to do with forward kinematics maybe allow animator to set fewer parameters or at least get a good first approximation What do we need skeleton with 123 dof joints solve from root position to end effector position arbitrary position constraints direction constraints joint limits techniques for handling underconstrained system adding constraints heuristics to push solution into right part of space optimization energy minimization Inverse Kinematics balance keep center of mass over support polygon control position vaulter s hands on line between shoulder and vault control compute knee angles that will give the runner the right leg length Inverse Kinematics 92 cos x2 yz L Li 2 LlL2 9 91 Lzsin92x L1 chos 92y Igsin92y L1 L2c0392X e f1x What makes IK hard equations hard to solve Xx YX ZX PX ax bx CX X Xy yy Zy py av by Cy dy XZ yZ ZZ pZ aZ bZ CZ dZ 0 0 0 1 0 0 0 1 abcd are functions of 9196 Xyzp are desired orientation position of end effector 12 equations 6 unknowns 9196 only 3 of the 9 rotation terms are independent non linear transcendantal equations What makes IK hard Redundancy a subspace 9X defined by 991en 8 9X if f9 X Add constraints to reduce redundancies Choose solution that is quotclosestquot to current configuration move outermost links the most energy minimization minimum time What makes IK hard singularities ill conditioned near singularities high state space velocities for lOW cartesian velocities What makes IK hard goal of quotnatural lookingquot motion minimum jerk equilibrium point trajectories The Iacobian f9 X X is of dimension n generally 6 9 is of dimension m of dof Jacobian is the n X m matrix relating differential changes of 9 dB to differential changes of X dX 19 d9 dX Where the ijth element of I is Jacobian maps velocities in state space to velocities in cartesian space Solutions no solution outside workspace too few dot multiple solutions redundancy single solution Methods closed form iterative IK and the Iacobian e f1x dXId9 d9 1 1 dX elm ek At 1 1 dX linearize about 9k Inverting the Iacobian I is n x m not square in general compute pseudo inverse Singularities cause the rank of the Iacobian to change Damped Least Squares find solution that minimizes III dX2 l2d92 tracking error joint velocities Non linear Optimization Zhao and Badler TOG 1994 solution is a local minima of some non linear function objective function constraints non linear optimization routine Objective Function position and orientation of end effector PX p X2 VXPm 2X p or just position or just orientation or aiming at Control Systems for Locomotion Where do control laws come from observation biomechanical literature optimization physical intuition add gures update Dynamic Models and Control Bruderlin Calvert 89 van de Panne Fiume and Vranesic 90 Raibert and Hodgins 91 Hodgins Wooten Brogan and O Brien 95 Generated Control Algorithms van de Panne and Fiume 93 Ngo and Marks 93 Sims 94 Tu and Terzopoulos 94 Hopper Dynamic Model 3 rigid bodies 2 or 4 controlled dof Control System state machine to structure control actions 777777 thrust lt compression islE 3AI Control of Hopping Velocity 777777 Body attitude i I Hopping height ii Velocity th th 12 ts Xd kxd de Xd 9 fxfh Body attitude 1 r k9 9d b d Hopping height f k1 1d bi id Control of Bipedal Running leg 1 active leg 2 acti R compressimh What to do with the idle leg mirror active leg held short swung forward held short swung forward How is walking different from running Double support in walking vs ight phase in running Energy transfer patterns support falling support rising correct pitch angle correct pitch angle lengthen swing leg shorten swing leg position swing leg position swing leg double support correct pitch angle thrust with rear leg Gait Transitions between walking and running walk gtrun extend leg support falling compression double support run gtwalk retract leg Gait Transitions between running and ipping Gymnastic Flips Quadruped Locomotion What else do we need Resistance to disturbances Strategies for error recovery Rough terrain locomotion More complicated mechanisms What else Rough Terrain Locomotion Dynamic Human Model 11 7 rigid bwdl caag 3CD mmm m cdm 39 handy wgmgm dems t ceg fmm b qnmmham ca data magg mcdl mmm mw f mcemigl MQM atced fmm pw ygwmza mand Dynamic Human Model external forces and torques enforce contact constraints Hierarchy of Control Laws State Machine for Running state machine to structure control actions compression decompression 1E1fKEP K t TM Hierarchy of Control Laws state machine control actions low level control Low level control proportional derivative servos C kp0d 0 kv T 2 Torque 661 6 2 Error between desired and actual joint angle 6 Angular velocity kpz Position control gain ICU 2 Velocity control gain Flight Duration 9d etd A9mom I ktfd Forward Velocity Xhh 12 tsic l k X Xd Agsm yhhz 12 tsy l k y yd Ground Speed Matching 1 Icp0d 9 Icv Dynamic Simulation point mass spring mass systems linkages of rigid bodies other physical phenomena aerodynamics uids fracture explosions Control Forward and Inverse Dynamics forward forces and accelerations torques inverse forward given forces and torques what is the motion inverse given prescribed motion what are the forces and torques Rigid Body Dynamics what can we simulate open loop lOintS rotary joints 123d telescoping joints 39 closed loop Forces Torques gravity wind collisions contact toning Dynamic Simulation physical description of objects gt equations of motion loop solve equations for accelerations integrate to find velocities and positions draw graphics Physical Description point mass translate in Xyz l y rigid body translate in Xyz rotate about Xyz articulated rigid body translate in Xyz J rotate about Xyz constraints wrt neighbors System Description mass center of mass moment of inertia Iz Iy IXIy112m3r2L2 Z Z x 2 X formula for simple solids Moment of Inertia of Polygonal Objects Lien and Kajiya 84 IEEE CGampA In 2D In 3D tetrahedrons Mirtich I Graphics Tools 1996 trlangles Algorithm to minimize numerical error and be efficient Code on the web Parallel Axis Theorem I I0 md2 I I 0


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