P11

补充1：Integration Methods Explained

Explicit Euler

By definition, the integral $\mathbf{x} (t) = \int \mathbf{v} (t) dt$ is the area. Many methods estimate the area as a box.

✅ 假设$\mathbf{x}$和$\mathbf{v}$都是一维的。速度的积分就是阴影区域的面积。

✅ 近似到一阶项，因此称为一阶方法。漏掉的高阶项就是误差。

P12

Implicit Euler

✅ 使用 $t_0$ 时刻的速度：显式积分
使用 $t_1$ 时刻的速度：隐式积分
两种方法都只能一阶近似

P13

Mid-Point

P14

比较与混合

By definition, the integral $\mathbf{x} (t)=∫\mathbf{v} (t) dt$ is the area. Many methods estimate the area as a box.

$\quad$

$\quad$

P15

$$\begin{cases} \mathbf{v} (t^{[1]})=\mathbf{v} (t^{[0]})+\mathbf{M} ^{−1}\int_{t^{[0]}}^{t^{[1]}} \mathbf{f} (\mathbf{x} (t), \mathbf{v} (t), t)dt\\ \mathbf{x} (t^{[1]})=\mathbf{x} (t^{[0]})+\int_{t^{[0]}}^{t^{[1]}} \mathbf{v} (t)dt \end{cases}$$

✅ 在当前应用场景中，使用前面方法的混合

P16

Leapfrog Integration

✅ 速度和位置是错开的。上下两种写法，在计算上是一样的。

In some literature, such a approach is called semi-implicit.

It has a funnier name: the leapfrog method.

P20

补充2：Rotation Representation

Rotation Represented by Matrix

• The matrix representation is widely used for rotational motion.

• It’s friendly for applying rotation to each vertex (by matrix-vector multiplication).

• But it is not suitable for dynamics:

  • It has too much redundancy: 9 elements but only 3 DoFs.
  • It is non-intuitive.
  • Defining its time derivative (rotational velocity) is also difficult.

P21

Rotation Represented by Euler Angles

• The Euler Angles representation is also popular, often in design and control.

• It is intuitive. It uses three axial rotations to represent one general rotation. Each axial rotation uses an angle.

• In Unity, the order is rotation-by-Z, rotation-by-X, then rotation-by-Y.

• But it is not suitable for dynamics either:

  • It can lose DoFs in certain statuses: gimbal lock.
  • Defining its time derivative (rotational velocity) is difficult.

P22

Gimbal Lock

The alignment of two or more axes results in a loss of rotational DoFs.

✅ 在某些特定的情况下，自由度降低了

P23

Rotation Represented by Quaternion

Introduction

In the complex system, two numbers represent a 2D point.

What about a “complex” system for 3D point? Quaternion! Four numbers represent a 3D point (with multiplication and division).

P24

Quaternion Arithematic

Let $\mathbf{q} = \begin{bmatrix} \mathbf{s} &\mathbf{v} \end{bmatrix}$ be a quaternion made of two parts: a scalar part $s$ and a 3D vector part $\mathbf{v}$, accounting for $\mathbf{ijk}$.

✅ 在有些库里面写作： $q = \begin{bmatrix} w & x & y &z \end{bmatrix}$，w为实数部分

$a\mathbf{q} =\begin{bmatrix} as &a\mathbf{v} \end{bmatrix}\quad$ Scalar-quaternion Multiplication

$\mathbf{q} _1±\mathbf{q} _2 =\begin{bmatrix} \mathbf{s}_1±\mathbf{s}_2 & \mathbf{v} _1 ± \mathbf{v} _2 \end{bmatrix}\quad\quad$ Addition/Subtraction

$\mathbf{q} _1×\mathbf{q} _2= \begin{bmatrix} \mathbf{s} _1\mathbf{s} _2−\mathbf{v} _1\cdot \mathbf{v} _2 & \mathbf{s} _1\mathbf{v} _2+\mathbf{s} _2\mathbf{v} _1+\mathbf{v} _1×\mathbf{v} _2 \end{bmatrix}\quad\quad$ Multiplication

$||\mathbf{q} ||=\sqrt{\mathbf{s^2+v\cdot v} } \quad\quad$Magnitude

$\quad$

P25

Rotation Represented by Quaternion

• To represent a rotation around $\mathbf{v}$ by angle $0$, we set the quaternion as:

  

• lt's very intuitive. lt's the built-in representation in Unity.

• Convertible to the matrix:

$$\mathbf{R}=\begin{bmatrix} s^2+x^2-y^2-z^2 & 2(xy-sz) & 2(xz+sy)\\ 2(xy+sz) & s^2-x^2+y^2-z^2 & 2(yz-sx) \\ 2(xz-sy) & 2(yz+sx) & s^2-x^2-y^2+z^2 \end{bmatrix}$$

本文出自CaterpillarStudyGroup，转载请注明出处。

https://caterpillarstudygroup.github.io/GAMES103_mdbook/