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| --- | ||
| layout: post | ||
| title: Robotics 4 - Kinematics | ||
| author: wichai | ||
| date: 2024-11-13 9:00 | ||
| categories: [Study, Master] | ||
| tags: [DU, Robotics] | ||
| mermaid: true | ||
| math: true | ||
| pin: false | ||
| --- | ||
|
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| From @VergilOP | ||
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| ## Lecture 4 - Kinematics | ||
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| ### Learning Objectives | ||
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| Objectives: | ||
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| 1. Spatial Description | ||
| 2. Transformation | ||
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| - Rotation | ||
| - Translation | ||
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| ### Spatial Description 空间描述 | ||
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| - Position of a Point 点的位置 | ||
| - With respect to a fixed origin O, the position of a point P is described by the vector OP(p) | ||
| 相对于固定原点 O，点 P 的位置由向量 OP(p) 描述 | ||
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| - Coordinate Frames: | ||
| - Rotation | ||
| - Translation | ||
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| - Rigid body configuration: | ||
| - Position: $^AP$ | ||
| - Orientation: $\{^AX_B, ^AY_B, ^AZ_B\}$ | ||
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| > These vectors describe rotation of {B} with respect to {A} | ||
| <img src="https://wichaiblog-1316355194.cos.ap-hongkong.myqcloud.com/image-20241128181058807.png" alt="image-20241128181058807" /> | ||
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| ### Transformation | ||
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| #### Rotation | ||
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|  | ||
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| - **Rotation Matrix:** | ||
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| **旋转矩阵（Rotation Matrix）**用于描述坐标系之间的旋转关系。假设有两个坐标系${A}$和${B}$，旋转矩阵$^A_BR$表示从坐标系${B}$到坐标系${A}$的旋转。 | ||
| $$ | ||
| ^A_BR | ||
| = \begin{bmatrix} | ||
| r_{11} & r_{12} & r_{13} \\ | ||
| r_{21} & r_{22} & r_{23} \\ | ||
| r_{31} & r_{32} & r_{33} | ||
| \end{bmatrix} | ||
| = \begin{bmatrix} ^A \hat{X}_B & ^A \hat{Y}_B & ^A \hat{Z}_B \end{bmatrix} | ||
| = \begin{bmatrix} {^B \hat{X}_A}^T \\ {^B \hat{Y}_A}^T \\ {^B \hat{Z}_A}^T \end{bmatrix} = {^B_A R}^T | ||
| = \begin{bmatrix} | ||
| \hat{X}_B \cdot \hat{X}_A & \hat{Y}_B \cdot \hat{X}_A & \hat{Z}_B \cdot \hat{X}_A\\ | ||
| \hat{X}_B \cdot \hat{Y}_A & \hat{Y}_B \cdot \hat{Y}_A & \hat{Z}_B \cdot \hat{Y}_A\\ | ||
| \hat{X}_B \cdot \hat{Z}_A & \hat{Y}_B \cdot \hat{Z}_A & \hat{Z}_B \cdot \hat{Z}_A | ||
| \end{bmatrix} | ||
| $$ | ||
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| 表达式 基向量组合形式 转置形式 点积形式 | ||
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| $^B \hat{X}_A$表示在坐标系${B}$中，坐标系${A}$的X轴单位向量。同理，其他项以此类推。 | ||
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| $\hat{X}_B$、$\hat{Y}_B$、$\hat{Z}_B$是坐标系${B}$的单位基向量，$\hat{X}_A$、$\hat{Y}_A$、$\hat{Z}_A$是坐标系${A}$的单位基向量。 | ||
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| 旋转矩阵是一个**正交矩阵（Orthogonal Matrix）**，其转置等于其逆矩阵。 | ||
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| > - Inverse of Rotation Matrix(Orthonormal Matrix) | ||
| > $$ | ||
| > ^A_BR^{-1} =\ ^B_AR =\ ^A_BR^T | ||
| > $$ | ||
| - State description: $^A\hat{X}_B = ^A_BR\ \ ^B\hat{X}_B$ | ||
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| $$ | ||
| ^A \hat{X}_B = {^A_B R} \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}\\ | ||
| ^A \hat{Y}_B = {^A_B R} \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}\\ | ||
| ^A \hat{Z}_B = {^A_B R} \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix} | ||
| $$ | ||
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| - Dot product: 通过点积，可以计算基向量在不同坐标系中的投影关系。 | ||
| $$ | ||
| ^A \hat{X}_B | ||
| = \begin{bmatrix} \hat{X}_B \cdot \hat{X}_A \\ \hat{X}_B \cdot \hat{Y}_A \\ \hat{X}_B \cdot \hat{Z}_A \end{bmatrix}\\ | ||
| ^A \hat{Y}_B | ||
| = \begin{bmatrix} \hat{Y}_B \cdot \hat{X}_A \\ \hat{Y}_B \cdot \hat{Y}_A \\ \hat{Y}_B \cdot \hat{Z}_A \end{bmatrix}\\ | ||
| ^A \hat{Z}_B | ||
| = \begin{bmatrix} \hat{Z}_B \cdot \hat{X}_A \\ \hat{Z}_B \cdot \hat{Y}_A \\ \hat{Z}_B \cdot \hat{Z}_A \end{bmatrix} | ||
| $$ | ||
| 其中，每个元素表示${B}$的基向量在${A}$的基向量方向上的投影。例如，$\hat{X}_B \cdot \hat{X}_A$表示${B}$的X轴在${A}$的X轴方向上的投影。 | ||
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| - Description of a Frame: | ||
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| - **描述一个坐标系需要知道其基向量和原点位置**： | ||
| - 坐标系${B}$在${A}$中的表示包括： | ||
| - 基向量：$^A \hat{X}_B$、$^A \hat{Y}_B$、$^A \hat{Z}_B$ | ||
| - 原点位置：$^A P_{B_{\text{org}}}$ | ||
| - Frame{B}: $^A \hat{X}_B, ^A \hat{Y}_B, ^A \hat{Z}_B$, $^AP_{Borg}$ | ||
| - 这里，$^A_BR$是从${B}$到${A}$的旋转矩阵，$^A P_{B_{\text{org}}}$是${B}$的原点在${A}$坐标系中的位置向量。 | ||
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| $$ | ||
| \{B\} = \{^A_BR\space\space^AP_{Borg}\} | ||
| $$ | ||
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| - Mapping: | ||
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| - Changing descriptions from frame to frame | ||
| - **映射**是指将一个向量从一个坐标系转换到另一个坐标系。 | ||
| - **旋转变换**： | ||
| - 当我们知道向量在${B}$坐标系中的表示$^B P$，想要得到它在${A}$坐标系中的表示$^A P$，可以使用旋转矩阵进行变换。 | ||
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| - Rotations | ||
|  | ||
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| - 假设有一个向量$P$，它在坐标系${B}$中的表示为$^B P$。我们想要计算它在坐标系${A}$中的表示$^A P$。 | ||
| - 首先，利用${B}$和${A}$的基向量之间的关系： | ||
| - If $P$ is in $\{B\}$: $^BP$ | ||
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| $$ | ||
| ^AP = \begin{bmatrix} | ||
| ^B \hat{X}_A. ^BP \\ ^B \hat{Y}_A. ^BP \\ ^B \hat{Z}_A. ^BP | ||
| \end{bmatrix} | ||
| = \begin{bmatrix} | ||
| ^B \hat{X}_A^T \\ ^B \hat{Y}_A^T \\ ^B \hat{Z}_A^T | ||
| \end{bmatrix} | ||
| \cdot\ ^BP | ||
| $$ | ||
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| $$ | ||
| ^AP =\ ^A_BR\ ^BP | ||
| $$ | ||
| - 这意味着，可以直接使用旋转矩阵$^A_BR$将向量从${B}$坐标系转换到${A}$坐标系。 | ||
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| #### Translation 平移 | ||
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|  | ||
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| $$ | ||
| ^AP_{OA} = ^AP_{OB} + ^AP_{BOrg} | ||
| $$ | ||
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| #### General Transformation | ||
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| #### <img src="https://wichaiblog-1316355194.cos.ap-hongkong.myqcloud.com/image-20241129103914524.png" alt="image-20241129103914524" style="zoom:50%;" /> | ||
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| $$ | ||
| ^AP =\ ^A_BR\ ^BP +\ ^AP_{Borg} \\ | ||
| \begin{bmatrix} | ||
| ^AP \\ | ||
| 1 | ||
| \end{bmatrix} | ||
| = \begin{bmatrix} | ||
| ^A_BR &\ ^AP_{Borg} \\ | ||
| 0\ 0\ 0 & 1 | ||
| \end{bmatrix} | ||
| = \begin{bmatrix} | ||
| ^BP\\ | ||
| 1 | ||
| \end{bmatrix} | ||
| $$ | ||
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| $^A P$：点$P$在坐标系${A}$中的表示。 | ||
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| $^B P$：点$P$在坐标系${B}$中的表示。 | ||
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| $^A_B R$：从${B}$到${A}$的旋转矩阵。 | ||
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| $^A P_{B_{\text{org}}}$：坐标系${B}$的原点在${A}$中的位置。 | ||
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| - Homogeneous Transformation: 齐次变换矩阵 | ||
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| 统一旋转和平移 | ||
| $$ | ||
| ^AP_{(4\times1)} =\ ^A_BT_{(4\times4)}\ ^BP_{(4\times1)} | ||
| $$ | ||
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| $^A_B T$是**齐次变换矩阵**，包括旋转和平移。 | ||
|
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| - General Operators: | ||
| $$ | ||
| P_2 = | ||
| \begin{bmatrix} | ||
| R_k(\theta) & Q \\ | ||
| 0\ 0\ 0 & 1 | ||
| \end{bmatrix}P_1 | ||
| $$ | ||
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| $$ | ||
| P_2 = T\ P_1 | ||
| $$ | ||
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| $P_1$：初始点的齐次坐标表示。 | ||
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| $P_2$：变换后的点的齐次坐标表示。 | ||
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| $R_k(\theta)$：绕轴$k$旋转$\theta$角度的旋转矩阵。 | ||
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| $Q$：平移向量。 | ||
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| - Inverse Transform 逆变换 | ||
| $$ | ||
| ^A_B T = \begin{bmatrix} | ||
| ^A_B R & ^A P_{Borg} \\ | ||
| 0\ 0\ 0 & 1 \end{bmatrix} | ||
| $$ | ||
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| $$ | ||
| ^A_B T^{-1} = ^B_A T = | ||
| \begin{bmatrix} | ||
| ^A_B R^T & -^A_B R^T \cdot\ ^AP_{Borg} \\ | ||
| 0\ 0\ 0 & 1 \end{bmatrix} | ||
| $$ | ||
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| - Homogeneous Transform Interpretations: | ||
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| - Description of a frame 坐标系的描述 | ||
|  | ||
| $$ | ||
| ^A_BT:\{B\} = \{^A_BR\ \ ^AP_{Borg}\} | ||
| $$ | ||
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| - Transform mapping 坐标的映射 | ||
|  | ||
| $$ | ||
| ^A_BT:\ ^BP \rarr\ ^AP | ||
| $$ | ||
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| - Transform operator 变换算子 | ||
|  | ||
| $$ | ||
| T: P_1 \rarr P_2 | ||
| $$ | ||
| $P_1$：初始点。 | ||
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| $P_2$：经过变换后的点。 | ||
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| - Compound Transformation:复合变换 | ||
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| <img src="https://wichaiblog-1316355194.cos.ap-hongkong.myqcloud.com/image-20241129105316388.png" alt="image-20241129105316388" style="zoom:50%;" /> | ||
| $$ | ||
| ^BP = ^B_C T \ C_P | ||
| $$ | ||
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| $$ | ||
| ^AP = ^A_B T \ B_P | ||
| $$ | ||
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| $$ | ||
| ^AP = ^A_B T \ ^B_C T \ C_P | ||
| $$ | ||
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| $$ | ||
| ^A_C T = ^A_B T \ ^B_C T | ||
| $$ | ||
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| $$ | ||
| ^A_C T = \begin{bmatrix} ^A_B R \ ^B_C R & ^A_B R \ ^B P_{Corg} + ^A P_{Borg} \\ 0\ 0\ 0 & 1 \end{bmatrix} | ||
| $$ | ||
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| - Transform Equation | ||
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| <img src="https://wichaiblog-1316355194.cos.ap-hongkong.myqcloud.com/image-20241129105332226.png" alt="image-20241129105332226" style="zoom:50%;" /> | ||
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| - 在多个坐标系之间的循环变换中，变换矩阵的乘积应等于单位矩阵： | ||
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| $$ | ||
| ^A_B T \ ^B_C T \ ^C_D T \ ^D_A T = I | ||
| $$ | ||
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| #### Representations | ||
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| - End-effector Configuration **末端执行器的齐次变换矩阵**： | ||
|  | ||
| $$ | ||
| ^B_ET: Position + Orientation | ||
| $$ | ||
| $^B_E T$表示末端执行器相对于基座坐标系${B}$的齐次变换矩阵，包含了位置和姿态的信息 | ||
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| - End-effector configuration parameters: | ||
| $$ | ||
| X = \begin{bmatrix} | ||
| X_P \\ | ||
| X_R | ||
| \end{bmatrix} | ||
| $$ | ||
| $X_P$：位置参数，表示末端执行器在空间中的位置。 | ||
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| **$X_R$**：姿态参数，表示末端执行器在空间中的方向或旋转。 | ||
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| - Position representation: | ||
|  | ||
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| - Cartesian: (x, y, z) 笛卡尔坐标系 | ||
| - Cylindrical: $(\rho, \theta, z)$ 圆柱坐标系 | ||
| - Spherical: $(r, \theta, \phi)$ 球坐标系 |
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