東京大学 情報理工学系研究科 創造情報学専攻 2012年8月実施 筆記試験 第3問

标签：

• Tokyo-University
• Control-Theory
• Inner-Product-and-Orthogonality
• Projection-Operator
• Rotation-Matrix-and-Axis-Angle

Author

itsuitsuki

Description

In kinematic calculations and visual computations in robotics, vector operations are expressed with matrices sometimes. Answer the following questions on inner-product, outer-product, projection and rotation of three dimensional vectors. $\boldsymbol{I}$ is the $3\times3$ identity matrix. The three-dimensional vectors $\boldsymbol{x}$, $\boldsymbol{a}$, $\boldsymbol{b}$ and $\boldsymbol{n}$ are $3\times1$ column vectors: $\boldsymbol{x}=\begin{bmatrix}x_x\\x_y\\x_z\end{bmatrix},\quad\boldsymbol{a}=\begin{bmatrix}a_x\\a_y\\a_z\end{bmatrix},\quad\boldsymbol{b}=\begin{bmatrix}b_x\\b_y\\b_z\end{bmatrix},\quad\boldsymbol{n}=\begin{bmatrix}n_x\\n_y\\n_z\end{bmatrix}.$ $\boldsymbol{x}^T$, which is a $1\times3$ row vector, shows the transpose of $\boldsymbol{x}$.

(1) On the inner product $(\boldsymbol{a},\boldsymbol{b})$ between vectors $\boldsymbol{a}$ and $\boldsymbol{b}$, describe i) the value of $(\boldsymbol{a},\boldsymbol{b})$, ii) a $3\times3$ matrix $\boldsymbol{A}$ which satisfies $(\boldsymbol{a},\boldsymbol{b})\boldsymbol{a}=\boldsymbol{A}\boldsymbol{b}$, and iii) $\boldsymbol{A}$ with vector $\boldsymbol{a}$ and its transpose $\boldsymbol{a}^T$.

(2) On the outer product $\boldsymbol{a}\times\boldsymbol{b}$ from $\boldsymbol{a}$ to $\boldsymbol{b}$, describe i) $3\times1$ expression of $\boldsymbol{a}\times\boldsymbol{b}$, ii) $3\times3$ expression of the matrix $\boldsymbol{A}$ which satisfies $\boldsymbol{a}\times\boldsymbol{b}=\boldsymbol{A}\boldsymbol{b}$, iii) the $3\times3$ matrix $\boldsymbol{Q}$ which satisfies $\boldsymbol{A}=\boldsymbol{a}\times\boldsymbol{Q}$ where $\boldsymbol{x}\times\boldsymbol{Q}$ between a vector $\boldsymbol{x}$ and a $3\times3$ matrix $\boldsymbol{Q}$ means a $3\times3$ matrix whose column vectors are three outer products from the vector $\boldsymbol{x}$ to each column vector in the matrix $\boldsymbol{Q}$ respectively.

(3) As Figure 1 shows, a vector $\boldsymbol{x}$ is vertically projected to a vector $\boldsymbol{y}$ on a plane whose normal vector is a unit vector $\boldsymbol{n}$. If the vector $\boldsymbol{y}$ is described as $\boldsymbol{y}=\boldsymbol{P}\boldsymbol{x}$, show that the matrix $\boldsymbol{P}$ becomes $\boldsymbol{P}=\boldsymbol{I}-\boldsymbol{n}\boldsymbol{n}^T$.

(4) Three rotational matrices $\boldsymbol{R}_x(\theta_x)$, $\boldsymbol{R}_y(\theta_y)$ and $\boldsymbol{R}_z(\theta_z)$ are rotational matrices which rotate a vector $\boldsymbol{x}$ around the X-axis, Y-axis and Z-axis with $\theta_x$, $\theta_y$ and $\theta_z$ respectively, where the direction of the rotation for plus is clock-wise around the axis from the origin to infinity.

i) Describe $3\times3$ expression of the matrix $\boldsymbol{R}_x(\theta_x)$, ii) As Figure 2 shows, the $3\times3$ matrix $\boldsymbol{R}_n(\theta_n)$ is defined as the rotation matrix around a unit orientation vector $\boldsymbol{n}$ with $\theta_n$. $\boldsymbol{R}_n(\theta_n)$ is described as $\boldsymbol{R}_n(\theta_n)=\boldsymbol{R}_x(-\alpha)\boldsymbol{R}_y(\beta)\boldsymbol{R}_z(\theta_n)\boldsymbol{R}_y(-\beta)\boldsymbol{R}_x(\alpha).$ Explain what the variables $\alpha$ and $\beta$ become and explain why the expression is satisfied.

题目描述

机器人运动学和视觉计算常用矩阵表示向量运算。回答三维向量的内积、外积、投影与旋转问题。令 $\boldsymbol I$ 为 $3\times3$ 单位矩阵，并令

$$\boldsymbol{x}=\begin{bmatrix}x_x\\x_y\\x_z\end{bmatrix},\quad \boldsymbol{a}=\begin{bmatrix}a_x\\a_y\\a_z\end{bmatrix},\quad \boldsymbol{b}=\begin{bmatrix}b_x\\b_y\\b_z\end{bmatrix},\quad \boldsymbol{n}=\begin{bmatrix}n_x\\n_y\\n_z\end{bmatrix}$$

均为 $3\times1$ 列向量；$\boldsymbol x^T$ 是其 $1\times3$ 转置行向量。

1. 对 $\boldsymbol a,\boldsymbol b$ 的内积 $(\boldsymbol a,\boldsymbol b)$，分别写出：a. 内积数值；b. 满足

   $$(\boldsymbol a,\boldsymbol b)\boldsymbol a=\boldsymbol A\boldsymbol b$$

   的 $3\times3$ 矩阵 $\boldsymbol A$；c. 用 $\boldsymbol a,\boldsymbol a^T$ 表示该 $\boldsymbol A$。

2. 对从 $\boldsymbol a$ 到 $\boldsymbol b$ 的外积 $\boldsymbol a\times\boldsymbol b$，分别写出：a. 其 $3\times1$ 分量表达式；b. 满足 $\boldsymbol a\times\boldsymbol b=\boldsymbol A\boldsymbol b$ 的 $3\times3$ 矩阵 $\boldsymbol A$；c. 满足 $\boldsymbol A=\boldsymbol a\times\boldsymbol Q$ 的 $3\times3$ 矩阵 $\boldsymbol Q$。这里 $\boldsymbol x\times\boldsymbol Q$ 定义为 $3\times3$ 矩阵，其每一列分别是 $\boldsymbol x$ 与 $\boldsymbol Q$ 对应列向量的外积。

3. 如图 1，把 $\boldsymbol x$ 正交投影到法向量为单位向量 $\boldsymbol n$ 的平面，得到 $\boldsymbol y$。若 $\boldsymbol y=\boldsymbol P\boldsymbol x$，证明

   $$\boldsymbol P=\boldsymbol I-\boldsymbol n\boldsymbol n^T.$$

4. $\boldsymbol R_x(\theta_x),\boldsymbol R_y(\theta_y),\boldsymbol R_z(\theta_z)$ 分别把向量绕 $X,Y,Z$ 轴旋转 $\theta_x,\theta_y,\theta_z$；从原点沿轴正方向看去，顺时针规定为正。

   1. 写出 $\boldsymbol R_x(\theta_x)$ 的 $3\times3$ 形式。

   2. 如图 2，$\boldsymbol R_n(\theta_n)$ 表示绕单位方向向量 $\boldsymbol n$ 旋转 $\theta_n$，且

      $$\boldsymbol R_n(\theta_n)= \boldsymbol R_x(-\alpha)\boldsymbol R_y(\beta)\boldsymbol R_z(\theta_n) \boldsymbol R_y(-\beta)\boldsymbol R_x(\alpha).$$

      求 $\alpha,\beta$，并解释该矩阵分解成立的原因。