Mesh Hierarchy Animation Part I – Rigid...

Sun-Jeong Kim

Mesh Hierarchy AnimationPart I – Rigid Meshes

7th Week, 2007

Game Programming II2

Objectives

To discover why mesh hierarchiesmesh hierarchies are important for object modeling

To understand the mathematics of mesh the mathematics of mesh hierarchy transformationshierarchy transformations

To become familiar with creating and traversing treetree--based mesh hierarchiesbased mesh hierarchies

To learn how to implement a robot arm and fictitious solar system using mesh hierarchies

To understand the basic idea of keyframeskeyframes and animation interpolation


Object Hierarchy (1)

Many objects are composed of parts, with a parent-child relationship

Hierarchy Transforms


Object Hierarchy (2)

A more complex hierarchy examplePelvis

Left Thigh

Left Calf

Left Foot

Right Thigh

Right Calf

Right Foot

Lower Spine

Upper Spine

Neck Left Clavicle

Left Upper Arm

Left Forearm

Left Hand

Right Clavicle

Right Upper Arm

Right Forearm

Right Hand


Robot Arm Demo


Mathematical Formulation (1)

A simple hierarchy

How do we correctly transform it to world space?Taking into consideration the transformations of its ancestors

Upper Arm(Bone 0)

Forearm(Bone 1)

Hand(Bone 2)

W

F0

F1 F2


Mathematical Formulation (2)

Matrix to transform the ith object in the arm hierarchy into world space

Ai: to-parent matrix to transform geometry from a child’s frame Fi into its parent frame Fi-1

W

F0

F1 F2

A0

A1

A2

A1A0

A2A1A0

011 AAAAM iii L−=


Bone Data Structure (1)

< Bone Mesh >


Bone Data Structure (2)


Building the Bone World Matrices


Animating the Bones


Rendering the Bones


Solar System Demo


Fictitious Solar System

Tree data structure for animated mesh hierarchy

0

1 2 3

4 5 6 7 8 9

[0] [1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]


Solar Object Data Structure (1)




Building the Solar Object World Matrices


Animating the Solar System


Keyframes and Animation (1)

KeyframeKeyframe – significant posepose of an object in the mesh hierarchy at some instance in time

Animations are prerecorded by a 3D animator or recorded from a motion capture system

Key1 taken attime t0 = 0.0s

Key2 taken attime t1 = 2.5s

Key3 taken attime t2 = 5.0s Key1 . . . . . . Key2 . . . . . . Key3

t0 = 0.0s t1 = 2.5s t2 = 5.0s

Calculate intermediate frames/poses



Each mesh in the hierarchy typically has several keyframes in an animation sequence

Pose is described by a rotation quaternion, scaling vector, and translation vector

Pose is described by a to-parent matrix

Keyframe interpolation

Key1 s = 0.0

Key2 s = 1.0

s = 0.167

s = 0.333

s = 0.50

s = 0.667

s = 0.833



Keyframe interpolation (cont’)RST-values

The to-parent matrix of a bone is decomposed into its rotation, scaling, and translation components

They are interpolated independently and then recombined into the to-parent matrix after interpolation

D3DX functions perform these interpolation techniques: D3DXVec3Lerp (linear interpolation) and D3DXQuaternionSlerp(spherical interpolation)



Pseudo codestruct Keyframe {

float time;D3DXQUATERNION R;D3DXVECTOR3 S;D3DXVECTOR3 T;

};

void interpolate( Keyframe &K0, Keyframe &K1, D3DXMATRIX &L ) {// Transform to [0, 1]float t0 = K0.time;float t1 = K1.time;float lerpTime = (t – t0) / (t1 – t0);

// Compute interpolated RST valuesD3DXVECTOR3 lerpedT;D3DXVECTOR3 lerpedS;D3DXQUATERNION lerpedR;



Pseudo code (cont’)D3DVec3Lerp( &lerpedT, &K0.T, &K1.T, lerpTime );D3DVec3Lerp( &lerpedS, &K0.S, &K1.S, lerpTime );D3DQuaternionSlerp( &lerpedR, &K0.R, &K1.R, lerpTime );

// Build and return the interpolated to-parent matrixD3DXMATRIX T, S, R;D3DXMatrixTranlation( &T, lerpedT.x, lerpedT.y, lerpedT.z );D3DXMatrixScaling( &S, lerpedS.x, lerpedS.y, lerpedS.z );D3DXRotationQuaternion( &R, &lerpedQ );

L = R * S * T;}


Exercises

Modify the Robot Arm demo to draw the bone hierarchy shown in the following figure. Allow the user to animate the bones as follows: rotate bone 0 on the y-axis; rotate bones 1 and 2 on the z-axis; and rotate bones 3, 4, 5, and 6 on the x-axis.

0

1

4

2

3

6

5

01 2

3

4

5

6

Mesh Hierarchy Animation Part I – Rigid...

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