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CSCE 590E Spring 2007 Animation By Jijun Tang Rendering Primitives Strips, Lists, Fans Indexed Primitives The Vertex Cache Quads and Point Sprites Strips, Lists, Fans Triangle strip 2 4 2 4 1 5 6 8 2 1 2 6 7 3 3 1 3 5 9 8 4 Triangle 1 3 list 3 4 5 2 7 4 5 5 6 6 1 Line list Line strip 6 Triangle fan Strips vs. Lists 32 triangles, 25 vertices 4 strips, 40 vertices 25 to 40 vertices is 60% extra data! Indexed Primitives Vertices stored in separate array No duplication of vertices Called a “vertex buffer” or “vertex array” Triangles hold indices, not vertices Index is just an integer Typically 16 bits Duplicating indices is cheap Indexes into vertex array Textures Texture Formats Texture Mapping Texture Filtering Rendering to Textures Texture Formats Textures made of texels Texels have R,G,B,A components Often do mean red, green, blue colors Really just a labelling convention Shader decides what the numbers “mean” Not all formats have all components Different formats have different bit widths for components Trade off storage space and speed for fidelity Common formats A8R8G8B8 (RGBA8): 32-bit RGB with Alpha 8 bits per comp, 32 bits total R5G6B5: 5 or 6 bits per comp, 16 bits total A32f: single 32-bit floating-point comp A16R16G16B16f: four 16-bit floats DXT1: compressed 4x4 RGB block 64 bits Storing 16 input pixels in 64 bits of output Consisting of two 16-bit R5G6B5 color values and a 4x4 two bit lookup table MIP Map 4x4 cube map 8x8 2D texture with (shown with sides mipmap chain expanded) Texture Filtering for Resize Point sampling enlarges without filtering When magnified, texels very obvious When minified, texture is “sparkly” Bilinear filtering Used to smooth textures when displayed larger or smaller than they actually are Blends edges of texels Texel only specifies color at centre Magnification looks better Minification still sparkles a lot Trilinear Filtering Trilinear can over-blur textures When triangles are edge-on to camera Especially roads and walls Anisotropic filtering solves this Takes multiple samples in one direction Averages them together Quite expensive in current hardware Lighting and Approaches Processes to determine the amount and direction of light incident on a surface how that light is absorbed, reemitted, and reflected which of those outgoing light rays eventually reach the eye Approaches: Forward tracing: trace every photon from light source Backward tracing: trace a photon backward from the eye Middle-out: compromise and trace the important rays Hemisphere lighting Three major lights: Sky is light blue Ground is dark green or brown Dot-product normal with “up vector” Blend between the two colors Good for brighter outdoor daylight scenes Example Lightmap Example Normal Mapping Example Specular Material Lighting Light bounces off surface How much light bounced into the eye? Other light did not hit eye – so not visible! Common model is “Blinn” lighting Surface made of “microfacets” They have random orientation With some type of distribution Example Environment Maps Blinn used for slightly rough materials Only models bright lights Light from normal objects is ignored Smooth surfaces can reflect everything No microfacets for smooth surfaces Only care about one source of light The one that reflects to hit the eye Example Character Animation What is animation? Animation is from the latin “anima” or soul To give motion Means to give life Anything you can do in your game to give it more “life” through motion (or lack of motion). Animation Example MSTS Overview Fundamental Concepts Animation Storage Playing Animations Blending Animations Motion Extraction Mesh Deformation Inverse Kinematics Attachments & Collision Detection Conclusions Animation Roles Programmer – loads information created by the animator and translates it into on screen action. Animator – Sets up the artwork. Provides motion information to the artwork. Different types of animation Particle effects Procedural / Physics “Hard” object animation (door, robot) “Soft” object animation (tree swaying in the wind, flag flapping the wind) Character animation 2D Versus 3D Animation Borrow from traditional 2D animation Image courtesy of George T. Henion. Understand the limitations of what can be done for real-time games Designing 3D motions to be viewed from more than one camera angle Pace motion to match game genre Animation terms frame – A image that is displayed on the screen, usually as part of a sequence. pose – a orientation of an objects or a hierarchy of objects that defines extreme or important motion. keyframe – a special frame that contains a pose. tween – the process of going “between” keyframes. secondary motion – An object motion that is the result of its connection or relationship with another object. Baking – setting every Nth frame as a key frame. Fundamental Problems Volume of data, processor limitations Mathematical complexity, especially for rotations. Translation of motion Fundamental Concepts Skeletal Hierarchy The Transform Euler Angles The 3x3 Matrix Quaternions Animation vs Deformation Models and Instances Animation Controls Skeletal Hierarchy The Skeleton is a tree of bones Often flattened to an array in practice Top bone in tree is the “root bone” May have multiple trees, so multiple roots Each bone has a transform Stored relative to its parent‟s transform Transforms are animated over time Tree structure is often called a “rig” Example The Transform “Transform” is the term for combined: Translation Rotation Scale Shear Can be represented as 4x3 or 4x4 matrix But usually store as components Non-identity scale and shear are rare Optimize code for common trans+rot case Examples Euler Angles Three rotations about three axes Intuitive meaning of values Euler Angles This means that we can represent an orientation with 3 numbers A sequence of rotations around principle axes is called an Euler Angle Sequence Assuming we limit ourselves to 3 rotations without successive rotations about the same axis, we could use any of the following 12 sequences: XYZ XZY XYX XZX YXZ YZX YXY YZY ZXY ZYX ZXZ ZYZ Using Euler Angles To use Euler angles, one must choose which of the 12 representations they want There may be some practical differences between them and the best sequence may depend on what exactly you are trying to accomplish Interpolating Euler Angles One can simply interpolate between the three values independently This will result in the interpolation following a different path depending on which of the 12 schemes you choose This may or may not be a problem, depending on your situation Interpolating near the „poles‟ can be problematic Note: when interpolating angles, remember to check for crossing the +180/-180 degree boundaries Problems Euler Angles Are Evil No standard choice or order of axes Singularity “poles” with infinite number of representations Interpolation of two rotations is hard Slow to turn into matrices Use matrix rotation Rotation Matrix 3x3 Matrix Rotation Easy to use Moderately intuitive Large memory size - 9 values Animation systems always low on memory Interpolation is hard Introduces scales and shears Need to re-orthonormalize matrices after Quaternions Quaternions are an interesting mathematical concept with a deep relationship with the foundations of algebra and number theory Invented by W.R.Hamilton in 1843 In practice, they are most useful to us as a means of representing orientations A quaternion has 4 components q q0 q1 q2 q3 Quaternions on Rotation Represents a rotation around an axis Four values <x,y,z,w> <x,y,z> is axis vector times sin(angle/2) w is cos(angle/2) No singularities But has dual coverage: Q same rotation as –Q This is useful in some cases! Interpolation is fast Quaternions (Imaginary Space) Quaternions are actually an extension to complex numbers Of the 4 components, one is a „real‟ scalar number, and the other 3 form a vector in imaginary ijk space! q q0 iq1 jq2 kq3 i j k ijk 1 2 2 2 i jk kj j ki ik k ij ji Quaternions (Scalar/Vector) Sometimes, they are written as the combination of a scalar value s and a vector value v q s, v where s q0 v q1 q2 q3 Unit Quaternions For convenience, we will use only unit length quaternions, as they will be sufficient for our purposes and make things a little easier q q0 q12 q2 q3 1 2 2 2 These correspond to the set of vectors that form the „surface‟ of a 4D hypersphere of radius 1 The „surface‟ is actually a 3D volume in 4D space, but it can sometimes be visualized as an extension to the concept of a 2D surface on a 3D sphere Quaternions as Rotations A quaternion can represent a rotation by an angle θ around a unit axis a: q cos a x sin a y sin a z sin 2 2 2 2 or q cos , a sin 2 2 If a is unit length, then q will be also Quaternions as Rotations q q q q q 2 0 2 1 2 2 2 3 cos 2 a sin 2 x 2 a sin 2 y 2 a sin 2 z 2 2 2 2 2 cos 2 2 sin 2 2 a 2 x a y a z2 2 cos sin a cos sin 2 2 2 2 2 2 2 2 2 1 1 Quaternion to Matrix To convert a quaternion to a rotation matrix: 1 2q2 2q3 2 2 2q1 q 2 2q 0 q 3 2q1 q 3 2q 0 q 2 2q1 q 2 2q 0 q 3 1 2q12 2q3 2q 2 q 3 2q 0 q1 2 2q1 q 3 2q 0 q 2 2q 2 q 3 2q 0 q1 1 2q1 2q2 2 2 Matrix to Quaternion Matrix to quaternion is doable It involves a few „if‟ statements, a square root, three divisions, and some other stuff Search online if interested Animation vs. Deformation Skeleton + bone transforms = “pose” Animation changes pose over time Knows nothing about vertices and meshes Done by “animation” system on CPU Deformation takes a pose, distorts the mesh for rendering Knows nothing about change over time Done by “rendering” system, often on GPU Pose Model Describes a single type of object Skeleton + rig One per object type Referenced by instances in a scene Usually also includes rendering data Mesh, textures, materials, etc Physics collision hulls, gameplay data, etc Instance A single entity in the game world References a model Holds current position & orientation (and gameplay state – health, ammo, etc) Has animations playing on it Stores a list of animation controls Animation Control Links an animation and an instance 1 control = 1 anim playing on 1 instance Holds current data of animation Current time Speed Weight Masks Looping state Animation Storage The Problem Decomposition Keyframes and Linear Interpolation Higher-Order Interpolation The Bezier Curve Non-Uniform Curves Looping Storage – The Problem 4x3 matrices, 60 per second is huge 200 bone character = 0.5Mb/sec Consoles have around 32-64Mb (Xbox and PS3 have larger, but still limited) Animation system gets maybe 25% PC has more memory But also higher quality requirements Decomposition Decompose 4x3 into components Translation (3 values) Rotation (4 values - quaternion) Scale (3 values) Skew (3 values) Most bones never scale & shear Many only have constant translation But human characters may have higher requirement Muscle move, smiling, etc. Cloth under winds Don‟t store constant values every frame, use index instead Keyframes Motion is usually smooth Only store every nth frame Store only “key frames” Linearly interpolate between keyframes Inbetweening or “tweening” Different anims require different rates Sleeping = low, running = high Choose rate carefully Key Frame Example 3D Canvas Linear Interpolation Higher-Order Interpolation Tweening uses linear interpolation Natural motions are not very linear Need lots of segments to approximate well So lots of keyframes Use a smooth curve to approximate Fewer segments for good approximation Fewer control points Bézier curve is very simple curve Bézier Curves (2D & 3D) Bézier curves can be thought of as a higher order extension of linear p1 interpolation p1 p1 p2 p3 p0 p0 p0 p2 The Bézier Curve (1-t)3F1+3t(1-t)2T1+3t2(1-t)T2+t3F2 T2 T1 t=1.0 F2 t=0.25 F1 t=0.0 The Bézier Curve (2) Quick to calculate Precise control over end tangents Smooth C0 and C1 continuity are easy to achieve C2 also possible, but not required here Requires three control points per curve (assume F2 is F1 of next segment) Far fewer segments than linear Bézier Variants Store 2F2-T2 instead of T2 Equals next segment T1 for smooth curves Store F1-T1 and T2-F2 vectors instead Same trick as above – reduces data stored Called a “Hermite” curve Catmull-Rom curve Passes through all control points Catmull-Rom Curve Defined by 4 points. Curve passes through middle 2 points. P = C3t3 + C2t2 + C1t + C0 C3 = -0.5 * P0 + 1.5 * P1 - 1.5 * P2 + 0.5 * P3 C2 = P0 - 2.5 * P1 + 2.0 * P2 - 0.5 * P3 C1 = -0.5 * P0 + 0.5 * P2 C0 = P1 Non-Uniform Curves Each segment stores a start time as well Time + control value(s) = “knot” Segments can be different durations Knots can be placed only where needed Allows perfect discontinuities Fewer knots in smooth parts of animation Add knots to guarantee curve values Transition points between animations “Golden poses” Looping and Continuity Ensure C0 and C1 for smooth motion At loop points At transition points Walk cycle to run cycle C1 requires both animations are playing at the same speed Reasonable requirement for anim system Playing Animations “Global time” is game-time Animation is stored in “local time” Animation starts at local time zero Speed is the ratio between the two Make sure animation system can change speed without changing current local time Usually stored in seconds Or can be in “frames” - 12, 24, 30, 60 per second Scrubbing Sample an animation at any local time Important ability for games Footstep planting Motion prediction AI action planning Starting a synchronized animation Walk to run transitions at any time Avoid delta-compression storage methods Very hard to scrub or play at variable speed Delta Compression Delta compression is a way of storing or transmitting data in the form of differences between sequential data rather than complete files. The differences are recorded in discrete files called deltas or diffs. Because changes are often small (only 2% total size on average), it can greatly reduce data redundancy. Collections of unique deltas are substantially more space-efficient than their non-encoded equivalents. Animation Blending The animation blending system allows a model to play more than one animation sequence at a time, while seamlessly blending the sequences Used to create sophisticated, life-like behavior Walking and smiling Running and shooting Blending Animations The Lerp Quaternion Blending Methods Multi-way Blending Bone Masks The Masked Lerp Hierarchical Blending The Lerp Foundation of all blending “Lerp”=Linear interpolation Blends A, B together by a scalar weight lerp (A, B, i) = iA + (1-i)B i is blend weight and usually goes from 0 to 1 Translation, scale, shear lerp are obvious Componentwise lerp Rotations are trickier – normalized quaternions is usually the best method. Quaternion Blending Normalizing lerp (nlerp) Lerp each component Normalize (can often be approximated) Follows shortest path Not constant velocity Multi-way-lerp is easy to do Very simple and fast Many others: Spherical lerp (slerp) Log-quaternion lerp (exp map) Which is the Best No perfect solution! Each missing one of the features All look identical for small interpolations This is the 99% case Blending very different animations looks bad whichever method you use Multi-way lerping is important So use cheapest - nlerp Multi-way Blending Can use nested lerps lerp (lerp (A, B, i), C, j) But n-1 weights - counterintuitive Order-dependent Weighted sum associates nicely (iA + jB + kC + …) / (i + j + k + … ) But no i value can result in 100% A More complex methods Less predictable and intuitive Can be expensive Bone Masks Some animations only affect some bones Wave animation only affects arm Walk affects legs strongly, arms weakly Arms swing unless waving or holding something Bone mask stores weight for each bone Multiplied by animation‟s overall weight Each bone has a different effective weight Each bone must be blended separately Bone weights are usually static Overall weight changes as character changes animations The Masked Lerp Two-way lerp using weights from a mask Each bone can be lerped differently Mask value of 1 means bone is 100% A Mask value of 0 means bone is 100% B Solves weighted-sum problem (no weight can give 100% A) No simple multi-way equivalent Just a single bone mask, but two animations Hierarchical Blending Combines all styles of blending A tree or directed graph of nodes Each leaf is an animation Each node is a style of blend Blends results of child nodes Construct programmatically at load time Evaluate with identical code each frame Avoids object-specific blending code Nodes with weights of zero not evaluated Motion Extraction Moving the Game Instance Linear Motion Extraction Composite Motion Extraction Variable Delta Extraction The Synthetic Root Bone Animation Without Rendering Moving the Game Instance Game instance is where the game thinks the object (character) is Usually just pos, orientation and bounding box Used for everything except rendering Collision detection Movement It‟s what the game is! Must move according to animations Linear Motion Extraction Find position on last frame of animation Subtract position on first frame of animation Divide by duration Subtract this motion from animation frames During animation playback, add this delta velocity to instance position Animation is preserved and instance moves Do same for orientation Problems Only approximates straight-line motion Position in middle of animation is wrong Midpoint of a jump is still on the ground! What if animation is interrupted? Instance will be in the wrong place Incorrect collision detection Purpose of a jump is to jump over things! Composite Motion Extraction Approximates motion with circular arc Pre-processing algorithm finds: Axis of rotation (vector) Speed of rotation (radians/sec) Linear speed along arc (metres/sec) Speed along axis of rotation (metres/sec) e.g. walking up a spiral staircase Benefits and Problems Very cheap to evaluate Low storage costs Approximates a lot of motions well Still too simple for some motions Mantling ledges Complex acrobatics Bouncing Variable Delta Extraction Uses root bone motion directly Sample root bone motion each frame Find delta from last frame Apply to instance pos+orn Root bone is ignored when rendering Instance pos+orn is the root bone Benefits Requires sampling the root bone More expensive than CME Can be significant with large worlds Use only if necessary, otherwise use CME Complete control over instance motion Uses existing animation code and data No “extraction” needed The Synthetic Root Bone All three methods use the root bone But what is the root bone? Where the character “thinks” they are Defined by animators and coders Does not match any physical bone Can be animated completely independently Therefore, “synthetic root bone” or SRB The Synthetic Root Bone (2) Acts as point of reference SRB is kept fixed between animations During transitions While blending Often at centre-of-mass at ground level Called the “ground shadow” But tricky when jumping or climbing – no ground! Or at pelvis level Does not rotate during walking, unlike real pelvis Or anywhere else that is convenient Animation Without Rendering Not all objects in the world are visible But all must move according to anims Make sure motion extraction and replay is independent of rendering Must run on all objects at all times Needs to be cheap! Use LME & CME when possible VDA when needed for complex animations Mesh Deformation Find Bones in World Space Find Delta from Rest Pose Deform Vertex Positions Deform Vertex Normals Find Bones in World Space Animation generates a “local pose” Hierarchy of bones Each relative to immediate parent Start at root Transform each bone by parent bone‟s world-space transform Descend tree recursively Now all bones have transforms in world space “World pose” Find Delta from Rest Pose Mesh is created in a pose Often the “da Vinci man” pose for humans Called the “rest pose” Must un-transform by that pose first Then transform by new pose Multiply new pose transforms by inverse of rest pose transforms Inverse of rest pose calculated at mesh load time Gives “delta” transform for each bone Deform Vertex Positions Deformation usually performed on GPU Delta transforms fed to GPU Usually stored in “constant” space Vertices each have n bones n is usually 4 4 bone indices 4 bone weights 0-1 Weights must sum to 1 Deform Vertex Positions (2) vec3 FinalPosition = {0,0,0}; for ( i = 0; i < 4; i++ ) { int BoneIndex = Vertex.Index[i]; float BoneWeight = Vertex.Weight[i]; FinalPosition += BoneWeight * Vertex.Position * PoseDelta[BoneIndex]); } Deform Vertex Normals Normals are done similarly to positions But use inverse transpose of delta transforms Translations are ignored For pure rotations, inverse(A)=transpose(A) So inverse(transpose(A)) = A For scale or shear, they are different Normals can use fewer bones per vertex Just one or two is common Inverse Kinematics FK & IK Single Bone IK Multi-Bone IK Cyclic Coordinate Descent Two-Bone IK IK by Interpolation FK & IK Most animation is “forward kinematics” Motion moves down skeletal hierarchy But there are feedback mechanisms Eyes track a fixed object while body moves Foot stays still on ground while walking Hand picks up cup from table This is “inverse kinematics” Motion moves back up skeletal hierarchy Single Bone IK Orient a bone in given direction Eyeballs Cameras Find desired aim vector Find current aim vector Find rotation from one to the other Cross-product gives axis Dot-product gives angle Transform object by that rotation Multi-Bone IK One bone must get to a target position Bone is called the “end effector” Can move some or all of its parents May be told which it should move first Move elbow before moving shoulders May be given joint constraints Cannot bend elbow backwards Cyclic Coordinate Descent Simple type of multi-bone IK Iterative Can be slow May not find best solution May not find any solution in complex cases But it is simple and versatile No precalculation or preprocessing needed Cyclic Coordinate Descent (2) Start at end effector Go up skeleton to next joint Move (usually rotate) joint to minimize distance between end effector and target Continue up skeleton one joint at a time If at root bone, start at end effector again Stop when end effector is “close enough” Or hit iteration count limit Cyclic Coordinate Descent (3) May take a lot of iterations Especially when joints are nearly straight and solution needs them bent e.g. a walking leg bending to go up a step 50 iterations is not uncommon! May not find the “right” answer Knee can try to bend in strange directions Two-Bone IK Direct method, not iterative Always finds correct solution If one exists Allows simple constraints Knees, elbows Restricted to two rigid bones with a rotation joint between them Knees, elbows! Can be used in a cyclic coordinate descent Two-Bone IK (2) Three joints must stay in user-specified plane e.g. knee may not move sideways Reduces 3D problem to a 2D one Both bones must remain same length Therefore, middle joint is at intersection of two circles Pick nearest solution to current pose Or one solution is disallowed Knees or elbows cannot bend backwards Two-Bone IK (3) Disallowed elbow position Shoulder Allowed Wrist elbow position IK by Interpolation Animator supplies multiple poses Each pose has a reference direction e.g. direction of aim of gun Game has a direction to aim in Blend poses together to achieve it Source poses can be realistic As long as interpolation makes sense Result looks far better than algorithmic IK with simple joint limits IK by Interpolation (2) Result aim point is inexact Blending two poses on complex skeletons does not give linear blend result Can iterate towards correct aim Can tweak aim with algorithmic IK But then need to fix up hands, eyes, head Can get rifle moving through body Attachments e.g. character holding a gun Gun is a separate mesh Attachment is bone in character‟s skeleton Represents root bone of gun Animate character Transform attachment bone to world space Move gun mesh to that pos+orn Attachments (2) e.g. person is hanging off bridge Attachment point is a bone in hand As with the gun example But here the person moves, not the bridge Find delta from root bone to attachment bone Find world transform of grip point on bridge Multiply by inverse of delta Finds position of root to keep hand gripping Collision Detection Most games just use bounding volume Some need perfect triangle collision Slow to test every triangle every frame Precalculate bounding box of each bone Transform by world pose transform Finds world-space bounding box Test to see if bbox was hit If it did, test the tris this bone influences Conclusions Use quaternions Matrices are too big, Eulers are too evil Memory use for animations is huge Use non-uniform spline curves Ability to scrub anims is important Multiple blending techniques Different methods for different places Blend graph simplifies code Conclusions (2) Motion extraction is tricky but essential Always running on all instances in world Trade off between cheap & accurate Use Synthetic Root Bone for precise control Deformation is really part of rendering Use graphics hardware where possible IK is much more than just IK algorithms Interaction between algorithms is key