C#和.Net框架

Animation / Simulation

• 动画是一种信息传递的工具
  • 美学经常比技术重要
• 是模型的延伸→连续性
  • Represent scene models as a function of time
• 输出：sequence of images that when viewed sequentially provide a sense of motion
  • 电影：24FPS
  • 视频：30FPS、29.994FPS
  • VR：90FPS （不晕的基础要求）

Keyframe animation关键帧动画

• Animator (e.g. lead animator) creates keyframes 关键帧
• Assistant (person or computer) creates in-between frames (“tweening”) 渐变帧

关键的技术难点 - Interpolation 插值

• Linear interpolation usually not good enough
• Recall splines for smooth / controllable interpolation

B样条……

Physical Simulation物理模拟

• 模拟、仿真：推导、实现公式，模拟出物体应该怎么变化
• 例子：布料模拟、流体模拟

质点弹簧系统 Mass Spring System: Example of Modeling a Dynamic System

• A Simple Idealized Spring
  • 没有初始长度
  • 随着拉力线性增长/缩短，线性系数是spring coefficient: stiffness
  • Force pulls points together
  • Strength proportional to displacement (Hooke’s Law)
  • 问题：长度会倾向于0

• Non-Zero Length Spring

  • 初始长度Rest length不为零

  • Problem: oscillates forever 永远震荡

    $$\boldsymbol{f}_{a \rightarrow b}=k_{s} \frac{\boldsymbol{b}-\boldsymbol{a}}{\|\boldsymbol{b}-\boldsymbol{a}\|}(\|\boldsymbol{b}-\boldsymbol{a}\|-l)$$
    • $\frac{\boldsymbol{b}-\boldsymbol{a}}{|\boldsymbol{b}-\boldsymbol{a}|}$表示受力方向
    • $|\boldsymbol{b}-\boldsymbol{a}|-l$表示弹簧拉伸长度
• Dot Notation for Derivatives：(用点表示导数)

$$\begin{aligned} &\boldsymbol{x}\\ &\dot{\boldsymbol{x}}=\boldsymbol{v}\\ &\ddot{\boldsymbol{x}}=\boldsymbol{a} \end{aligned}$$

• Introducing Energy Loss

  • Simple motion damping 阻尼(对弹簧全局作用)

    $$\boldsymbol{f}=-k_{d} \dot{\boldsymbol{b}}$$

  • Behaves like viscous drag on

  • Slows down motion in the direction of velocity
  • $k_d$ is a damping coefficient
  • 问题：Slows down all motion
    • Want a rusty spring’s oscillations to slow down, but should it also fall to the ground more slowly? 跟全局速度挂钩
    • 无法表示弹簧内部的损耗

• Internal Damping for Spring

  

  • 相对运动越快,f就越大,所以跟局部加速度有关

  • 注意这里一定要将质量在相对加速度投影到弹簧ab方向,才是影响弹簧的力,比如旋转的时候加速度方向和弹簧方向垂直,不影响.

  • Viscous drag only on change in spring length

    • 粘性阻力仅在弹簧长度变化时
    • Won’t slow group motion for the spring system (e.g. global translation or rotation of the group)
      • 弹簧系统不会减慢群组运动（例如，群组的整体平移或旋转）
  • Note: This is only one specific type of damping 只是一种阻尼的近似

Structures from Springs

• 通过质点弹簧系统来进行布料模拟
  Step 1: Sheets

• This structure will not resist shearing切变会露馅
• This structure will not resist out-of-plane bending…

Step 2: 增加斜向方向弹簧

• This structure will resist shearing but has anisotropic bias 各向异性
• This structure will not resist out-of-plane bending either…

Step 3: 增加双斜向方向弹簧

• This structure will resist shearing. Less directional bias.
• This structure will not resist out-of-plane bending either… 弯折

Step 4: 增加横向和纵向 （skip connection）

• This structure will resist shearing. Less directional bias.
• This structure will resist out-of-plane bending （横向和纵向的弹簧力比斜向的弱很多）(对折)

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有限元方法(FEM (Finite Element Method) Instead of Springs)

• 车辆碰撞

力传导扩散适合用有限元方法建模做

Particle Systems

• 建模定义很多粒子
• 每个粒子有自己的属性

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• Model dynamical systems as collections of large numbers of particles
• Each particle’s motion is defined by a set of physical (or non-physical) forces
• Popular technique in graphics and games
  • Easy to understand, implement
  • Scalable: fewer particles for speed, more for higher complexity

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• Challenges
  • May need many particles (e.g. fluids)
  • May need acceleration structures (e.g. to find nearest particles for interactions)

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• 简易算法

For each frame in animation
  [If needed] Remove dead particles
  Calculate forces on each particle
  Update each particle’s position and velocity
  [If needed] Create new particles
  Render particles

• 定义个体和群体之间的关系

Particle System Forces

Attraction and repulsion forces

• Gravity, electromagnetism, …

• Springs, propulsion, …

Damping forces

• Friction, air drag, viscosity, …

Collisions

• Walls, containers, fixed objects, …

• Dynamic objects, character body parts, …

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Example: Simulated Flocking as an ODE(常微分方程)

• 定义鸟儿之间交互的规则：个体对群体的观察
• Model each bird as a particle Subject to very simple forces:
• attraction to center of neighbors
• repulsion from individual neighbors
• alignment toward average trajectory of neighbors Simulate evolution of large particle system numerically Emergent complex behavior (also seen in fish, bees, …)

Example: Molecular Dynamics

Example: Crowds + “Rock” Dynamics

Kinematics

• 运动学：正向和反向

Forward Kinematics 正向运动学

明确骨骼之间的运动关系→计算出各个部位的位置

Articulated skeleton

• Topology (what’s connected to what)
• Geometric relations from joints
• Tree structure (in absence of loops)

Joint types

• Pin (1D rotation)
• Ball (2D rotation)
• Prismatic joint (translation)

Strengths

• Direct control is convenient
• Implementation is straightforward

Weaknesses

• Animation may be inconsistent with physics
• Time consuming for artists

Inverse Kinematics 逆运动学

• 限制各个部位（通常只有终端）的位置、限制骨骼的运动方式→计算骨骼的运动
• 方便控制形体整体形状
• 解特别复杂，可能并不唯一,也可能无解

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解法：随机化算法（优化方法，梯度下降）

Numerical solution to general N-link IK problem

• Choose an initial configuration

• Define an error metric (e.g. square of distance between goal and current position)

• Compute gradient of error as function of configuration

• Apply gradient descent (or Newton’s method, or other optimization procedure)

例子：Style-Based IK

Rigging

通过控制点对三维物体进行控制(比如改变脸部表情,手势等)

• Rigging is a set of higher level controls on a character that allow more rapid & intuitive modification of pose, deformations, expression, etc.

• Important

  • Like strings on a puppet
  • Captures all meaningful character changes
  • Varies from character to character
• Expensive to create
  • Manual effort 定控制点，拉控制点（应该怎么定、应该怎么拉 → 动画师）
  • Requires both artistic and technical training

Blend Shapes 控制点间的位置插值计算

• Instead of skeleton, interpolate directly between surfaces
  • E.g., model a collection of facial expressions:
• Simplest scheme: take linear combination of vertex positions
• Spline used to control choice of weights over time

Motion Capture

• 真人控制点反映到虚拟角色中去，需要建立真实和虚拟的联系

Data-driven approach to creating animation sequences

• Record real-world performances (e.g. person executing an activity)
• Extract pose as a function of time from the data collected

Strengths

• Can capture large amounts of real data quickly

• Realism can be high

Weaknesses

• Complex and costly set-ups 复杂、花钱

• Captured animation may not meet artistic needs, requiring alterations 不符合艺术家要求，不可能实现的动作

• 捕捉条件限制

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不同的捕捉方法：

• Optical (More on following slides)
  • Markers on subject
  • Positions by triangulation from multiple cameras
  • 8+ cameras, 240 Hz, occlusions are difficult
• Magnetic Sense magnetic fields to infer position / orientation. Tethered.
• Mechanical Measure joint angles directly. Restricts motion.

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Challenges of Facial Animation

• Uncanny valley
  • In robotics and graphics
  • As artificial character appearance approaches human realism, our emotional response goes negative, until it achieves a sufficiently convincing level of realism in expression

动画的制作流程 The Production Pipeline

Single Particle Simulation

• 假设粒子在的运动由速度场决定
  • 速度场是一个关于位置和时间的函数 $v(x,t)$
• 对于粒子我们知道在时间$t_{0}$的速度$v_{t0}$和位置$x_{t0}$
  • 我们要求的是任意时间$t_{1}$的位置$x_{t1}$

Ordinary Differential Equation (ODE) 常微分方程

计算速度场内粒子的位置需要计算一阶常微分方程：

$$\frac{d x}{d t}=\dot{x}=v(x, t)$$

解一阶常微分方程：

• 连续：积分

• 离散：Euler’s Method欧拉方法

  • Explicit Euler method （forward 前向、显式）

    • 始终用前一帧的状态来更新后一帧

      $$\begin{array}{l} \boldsymbol{x}^{t+\Delta t}=\boldsymbol{x}^{t}+\Delta t \dot{\boldsymbol{x}}^{t} \\ \dot{\boldsymbol{x}}^{t+\Delta t}=\dot{\boldsymbol{x}}^{t}+\Delta t \ddot{\boldsymbol{x}}^{t} \end{array}$$

    • Simple iterative method

    • Commonly used
  • 问题:
    • Very inaccurate 很不准确:步长越大越不准确
      • With numerical integration, errors accumulate
      • Euler integration is particularly bad
    • Most often goes unstable 不稳定
      • 容易出现正反馈，离正确结果越来越远
        

Instability and improvements

Solving by numerical integration with finite differences leads to two problems:

Errors 误差 不是特别大的问题

• Errors at each time step accumulate. Accuracy decreases as simulation proceeds
• Accuracy may not be critical in graphics applications

Instability 不稳定性 很要命！

• Errors can compound, causing the simulation to diverge even when the underlying system does not 收敛很重要！
• Lack of stability is a fundamental problem in simulation, and cannot be ignored

Combating Instability

Midpoint method

• 用一次欧拉方法得到a点
• 取起点到a点的终点b,得到对应的速度
• 在起始点应用b点的速度进行欧拉方法,得到c点

Adaptive Step Size

• 用一次欧拉方法得到$X_{T}$点
• 在中点处再用欧拉方法得到$X_{T/2}$点
• 判断两个点之间的距离,超过一定距离递归的缩小步长

Implicit Eulr Method

• Use the velocity at the next time step (hard)

$$\begin{array}{l} \boldsymbol{x}^{t+\Delta t}=\boldsymbol{x}^{t}+\Delta t \dot{\boldsymbol{x}}^{t+\Delta t} \\ \dot{\boldsymbol{x}}^{t+\Delta t}=\dot{\boldsymbol{x}}^{t}+\Delta t \ddot{\boldsymbol{x}}^{t+\Delta t} \end{array}$$

• 是一个方程组，有三个未知数，只能假设一个已知（猜）
• Use root-finding algorithm, e.g. Newton’s method
• Offers much better stability

• Implicit Euler has order 1, which means that

  • Local truncation error: O(h^2) and

  • Global truncation error: O(h) (h is the step, i.e. ∆t)

    O(h)的理解

  • If we halve h, we can expect the error to halve as well

  • 阶数越高越好，减小步长的情况下降低更快

Runge-Kutta Families

• A family of advanced methods for solving ODEs
• Especially good at dealing with non-linearity
• It’s order-four version is the most widely used, a.k.a. RK4

更多：Numerical Analysis 对图形学有用 （其他的：信号处理）

Position-Based / Verlet Intergation

• Constrain positions and velocities of particles after time step

假设弹簧无限大

Idea:

• After modified Euler forward-step, constrain positions of particles to prevent divergent, unstable behavior
• Use constrained positions to calculate velocity
• Both of these ideas will dissipate energy, stabilize

Pros / cons

• Fast and simple
• Not physically based, dissipates energy (error)

Rigid body simulation

不会形变

Simple case

• Similar to simulating a particle
• Just consider a bit more properties 拓展

Fluid Simulation

A Simple Position-Based Method

模拟只需要输出物体的位置，其他的就是渲染的问题了

Key idea

• Assuming water is composed of small rigid-body spheres 水是由小球组成的
• Assuming the water cannot be compressed (i.e. const. density) 不可压缩
• So, as long as the density changes somewhere, it should be “corrected” via changing the positions of particles 密度有变化，就要想着改回去，移动小球位置

• You need to know the gradient of the density anywhere w.r.t. each particle’s position

  一个小球位置的变化对其周围密度的影响

• Update? Just gradient descent! 梯度下降

非物理

流体模拟中两种不同的思路：Eulerian vs. Lagrangian

Lagrangian：质点法，以每个元素为单位模拟

Eulerian：网格法，以空间为单位分割模拟

Material Point Method (MPM)

Hybrid, combining Eulerian and Lagrangian views

• Lagrangian: consider particles carrying material properties 物质由粒子组成，粒子有属性
• Eulerian: use a grid to do numerical integration 网格上计算如何运动
• Interaction: particles transfer properties to the grid, grid performs update, then interpolate back to particles 网格属性写回网格内粒子上