## 2013年9月15日日曜日

### Kinematics 〜 Implementation 〜

in Japanese

#### Introduction

In the previous page, I described a brief explanation on the kinematics. Here I show you my implementation of the algorithm of a six-axis-arm system in C++.

#### Demo

A green sphere indicates a target position. When the end point of the arm arrives at it, the target is moved to the next position.

## Development Environment

1. Mac OS X 10.8.4
2. Processor：3.06 GHz Intel Core 2 Duo
3. Memory：4GB
4. Xcode4.6.2 with Apple LLVM 4.2（C++ Language Dialect → C++11, C++ Standard Library → libc++）
5. boost-1.54.0（built by Apple LLVM 4.2 compiler）

#### Source

Here is my source code.

#### Usage

A file robot.dat passed as an argument determines a form of the arm system. The format of the file is expressed in 1. Using keyboard,
1. 'c' starts the program,
2. 'b' inverts the movement of the target sphere,
3. 'a' restores the movement,
4. and escape stops the program.
The end point of the arm system gradually gives rise to a distortion from the target point. We have to introduce the feedback function.

## References

1. OpenGL 3Dグラフィックス入門 第2版 三浦憲二郎 朝倉書店 (In Japanese)

### Kinematics 〜 Theory 〜

in Japanese

#### Introduction

In this page, I describe a brief explanation on the kinematics used in the robotics. See also the next page as to the implementation of a simple six-axis-arm system in C++.

#### Forward Kinematics

Let us consider the 3-dimensional homogeneous coordinate system. In the system, the rotation matrices around x, y, and z axises are given by \begin{equation} R_{x}(\theta) = \left( \begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & \cos{\theta} & -\sin{\theta} & 0 \\ 0 & \sin{\theta} & \cos{\theta} & 0 \\ 0 & 0 & 0 & 1 \end{array} \right), \end{equation} \begin{equation} R_{y}(\theta) = \left( \begin{array}{cccc} \cos{\theta} & 0 & \sin{\theta} & 0 \\ 0 & 1 & 0 & 0 \\ -\sin{\theta} & 0 & \cos{\theta} & 0 \\ 0 & 0 & 0 & 1 \end{array} \right), \end{equation} and \begin{equation} R_{z}(\theta) = \left( \begin{array}{cccc} \cos{\theta} & -\sin{\theta} & 0 & 0 \\ \sin{\theta} & \cos{\theta} & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array} \right). \end{equation} Moreover, the translation matrix $L(\vec{l})$ with a directional vector $\vec{l}$ is obtained by \begin{equation} L(\vec{l}) = \left( \begin{array}{cccc} 1 & 0 & 0 & l_x \\ 0 & 1 & 0 & l_y \\ 0 & 0 & 1 & l_z \\ 0 & 0 & 0 & 1 \end{array} \right). \end{equation} Suppose that we have an initial state of a three-axis-arm system shown in the below figure. In the figure, $n_i$ and $l_i$ indicate a joint and a length of each component, respectively. $n_1$ rotates $\theta_1$ in counterclockwise direction around $x$ axis, $n_2$ rotates $\theta_2$ in counterclockwise direction around $y$ axis, and $n_3$ rotates $\theta_3$ in counterclockwise direction around $x$ axis. $\vec{l}_1=(0,0,l_1)$, $\vec{l}_2=(0,0,l_2)$, and $\vec{l}_3=(0,0,l_3)$. A position vector of an end point $n_4$ takes the form \begin{equation} \vec{p}_{4} = R_x(\theta_1) L(\vec{l}_1) R_y(\theta_2) L(\vec{l}_2) R_x(\theta_3) L(\vec{l}_3) \left( \begin{array}{c} 0 \\ 0 \\ 0 \\ 1 \end{array} \right). \label{sample-case} \end{equation} It must be noted that parameters $\theta_i$ and $\vec{l}_i$ are defined in the local coordinate system of the $i$-th joint. We can expand eq.(\ref{sample-case}) to the case of $n$-axis-arm system as \begin{equation} \vec{p} = R_{a_{1}}(\theta_1) L(\vec{l}_1) R_{a_{2}}(\theta_2) L(\vec{l}_2) \cdots R_{a_{n}}(\theta_n) L(\vec{l}_n) \left( \begin{array}{c} 0 \\ 0 \\ 0 \\ 1 \end{array} \right), \end{equation} where $a_{i}\in \{x,y,z\}$.

#### Inverse Kinematcis

Setting lengths to constant values, the vector $\vec{p}$ becomes a function with $n$ angles in the form \begin{equation} \vec{p}= \vec{f}(\theta_1,\theta_2,\cdots,\theta_n). \end{equation} Its deviation with respect to $\{\theta_1,\theta_2,\cdots,\theta_n\}$ is written as \begin{eqnarray} \delta\vec{p} &=& \vec{f}(\theta_1+\delta\theta_1,\theta_2+\delta\theta_2,\cdots,\theta_n+\delta\theta_n) - \vec{f}(\theta_1,\theta_2,\cdots,\theta_n)\nonumber \\ &=&\sum_{i=1}^{n}\;\frac{\partial\vec{f}}{\partial\theta_i}\;\delta\theta_i + \frac{1}{2}\sum_{i=1}^{n}\sum_{j=1}^{n}\;\frac{\partial^2\vec{f}}{\partial\theta_i\partial\theta_j}\;\delta\theta_i\;\delta\theta_j +\cdots. \end{eqnarray} Ignoring the terms more than first derivative, we can obtain the form \begin{eqnarray} \left( \begin{array}{c} \delta p_x \\ \delta p_y \\ \delta p_z \end{array} \right) &=& \left(\frac{\partial\vec{f}}{\partial\theta_1},\frac{\partial\vec{f}}{\partial\theta_2},\cdots,\frac{\partial\vec{f}}{\partial\theta_n}\right) \left( \begin{array}{c} \delta \theta_1 \\ \delta \theta_2 \\ \vdots \\ \delta \theta_n \end{array} \right) \nonumber \\ &=& \left( \begin{array}{cccc} \frac{\partial f_1}{\partial \theta_1} & \frac{\partial f_1}{\partial \theta_2} & \cdots & \frac{\partial f_1}{\partial \theta_n} \\ \frac{\partial f_2}{\partial \theta_1} & \frac{\partial f_2}{\partial \theta_2} & \cdots & \frac{\partial f_2}{\partial \theta_n} \\ \frac{\partial f_3}{\partial \theta_1} & \frac{\partial f_3}{\partial \theta_2} & \cdots & \frac{\partial f_3}{\partial \theta_n} \end{array} \right) \left( \begin{array}{c} \delta \theta_1 \\ \delta \theta_2 \\ \vdots \\ \delta \theta_n \end{array} \right) \nonumber \\ &\equiv& J \left( \begin{array}{c} \delta \theta_1 \\ \delta \theta_2 \\ \vdots \\ \delta \theta_n \end{array} \right), \end{eqnarray} where $J$ is called the Jacobian matrix. In this case, it is $3 \times n$ matrix. A partial differential $\frac{\partial \vec{f}}{\partial \theta_i}$ is calculated as \begin{equation} \frac{\partial \vec{f}}{\partial \theta_i} = R_{a_{1}}(\theta_1) L(\vec{l}_1) \cdots \frac{dR_{a_{i}}(\theta_i)}{d\theta_i} L(\vec{l}_i) \cdots R_{a_{n}}(\theta_n) L(\vec{l}_n) \left( \begin{array}{c} 0 \\ 0 \\ 0 \\ 1 \end{array} \right). \end{equation} By introducing the inverse matrix $J^{-1}$, we can obtain the deviation vector $\delta \vec{\theta}$ of angles as \begin{equation} \delta \vec{\theta} = J^{-1}\;\delta\vec{p}. \end{equation} When ${\rm rank}(J)>3$, the inverse matrix $J^{-1}$ is not uniquely determined. In other words, in a case where the number of joints are greater than 3, the end position ($\vec{p}$) is achieved by different postures. The typical method to determine a unique posture is to use a pseudoinverse matrix $J^{\#}$ as \begin{equation} \delta \vec{\theta} = J^{\#} \delta\vec{p}, \end{equation} where \begin{equation} J^{\#}=J^{T}\;(J\;J^{T})^{-1}. \end{equation} The procedure to solve the inverse kinematics problem is as follows:
1. Suppose that a target position of the end point is $\vec{p}_{\rm G}$.
2. Calculate a displacement $\vec{d} = \vec{p}_{\rm G}-\vec{p}$ where $\vec{p}$ indicates the current positoin of the end point. Using $\vec{d}$, we make a small deviation $\delta \vec{p}=\alpha\;\vec{d}/|\vec{d}|$ where $\alpha$ is an appropriate positive value.
3. Using current angles, we calculate $J^{\#}$.
4. Calculate $\delta \vec{\theta}=J^{\#} \delta\vec{p}$.
5. Update the quantities, $\vec{\theta} \leftarrow \vec{\theta} + \delta \vec{\theta}, \vec{p} \leftarrow \vec{p} + \delta \vec{p}$.
6. After updating them, if the displacement $|\vec{d}| = |\vec{p}_{\rm G}-\vec{p}|$ is not in an acceptable error range $\epsilon$, return to 2.

#### References

1. Forward Kinematics (in Japanese)
2. Inverse Kinematics by means of Jacobian (in Japanese)
3. Kinematics Problem of Robot (in Japanese)

## 2013年9月7日土曜日

### 運動学　〜 実装 〜

in English

#### はじめに

先のページで解説したアルゴリズムをC++11で実装した。

## 開発環境

1. Mac OS X 10.8.4
2. プロセッサ：3.06 GHz Intel Core 2 Duo
3. メモリ：4GB
4. Xcode4.6.2 with Apple LLVM 4.2（C++ Language Dialect → C++11, C++ Standard Library → libc++）
5. boost-1.54.0（Apple LLVM 4.2でコンパイルしたもの。）

ソースはここ

#### 実行方法

1. 'c'を入力してスタート
2. 'b'を入力すると逆向き
3. 'a'を入力すると元の向き
4. escapeキーで終了

## 参考文献

1. OpenGL 3Dグラフィックス入門 第2版 三浦憲二郎 朝倉書店