- [rigidbody transformation](#rigidbody transformation)
- [FK](# forward kinematics)
- [IK](# inverse kinematics)
旋转运动
根据欧拉定理，任意三维空间旋转运动可以表示为绕某一单位轴$w\in \mathcal{R}^3$转动角度$\theta$,则旋转矩阵可以表示为矩阵指数的形式：
$$R=e^{\hat{w}\theta} \in \mathcal{R}^{3\times 3} \
\hat{w}=\begin{bmatrix}0 &-w_3 & w_2 \w_3 & 0 & -w_1 \-w_2 & w_1 & 0 \end{bmatrix}$$
$$so(3)与SO(3)的指数映射\text{(Exponential map)}关系 {\begin{array}{lr}
\hat{w}\in so(3)\in \mathcal{R}^{3\times 3} \
\quad \
\theta \in \mathcal{R}
\end{array} \Rightarrow e^{\hat{w}\theta}\in SO(3)$$
angles:
rotation angle: $[n]\times=\frac{R-R^T}{2sin(\theta)}$, $n=[-[n]\times(2,3),[n]\times(1,3),-[n]\times(1,2)]$
In unit quaternions(四元数): q=$(cos(\theta/2),\omega sin(\theta/2))$
刚体运动
与旋转相似，根据Chasels定理，任意刚体运动均可以通过绕一轴的运动加上平行于该轴的移动实现: $$T=e^{\hat{\xi}\theta} \in \mathcal{R}^{4\times 4} \ \hat{\xi}=\begin{bmatrix} \hat{w} & \upsilon \ 0& 0 \end{bmatrix} \ e^{\hat{\xi}\theta}=\begin{bmatrix} e^{\hat{w}\theta}& (I-e^{\hat{w}\theta})(w\times \upsilon)+ww^T\upsilon\theta\0 &1 \end{bmatrix}$$
- for revolute joint:
$\theta_i \in S^1$ ,unit circle in the plane - for prismatic joint:
$\theta_i \in \mathcal{R}$ -
$T^p$ to denote the p-torus, defined to be the Cartesian product of p copies of$S^1$ :$$T^p=S^1\times S^1\times \cdots \times S^1$$ - joint space(configuration space) of a manipulator with p revolute joints and r prismatic joints:
$Q=T^p \times R^ r$ - the forward kinematics map:
$g_{st}$ , t-tool frame and s-base frame:$$g_{st}(\theta)=g_{sl_1}(\theta_1)g_{l_1l_2}(\theta_2)\cdots g_{l_nt}$$
指数积公式
If
The product of exponential formula(POE): $$g_{st}(\theta)=e^{\hat{\xi}_1\theta_1}e^{\hat{\xi}_2\theta_2}\dots e^{\hat{\xi}n\theta_n}g{ab}(0)$$
Generalize this procedure:
- Define the reference configuration(初始位置) of the manipulator to be the configuration corresponding to
$\theta=0$ . - For each joint, construct a twist
$\xi_i$ at$g_{st}(0)$ -
$\omega_i \in \mathcal{R}^3$ ,unit vector in the direction of th twist axis -
$q_i \in \mathcal{R}^3$ ,any point on the axis(determine the direction) -
$v_i \in \mathcal{R}^3$ ,unit vector pointing in the direction of translation - all vectors are specified relative to the base coordinate frame $$\begin{align} \text{for revolute } \xi_i&=\begin{bmatrix}-\omega_i\times q_i \ \omega_i \end{bmatrix} or \begin{bmatrix}q_i\times \omega_i \ \omega_i \end{bmatrix} \ \text{for prismatic } \xi_i&=\begin{bmatrix}v_i \ 0 \end{bmatrix} \end{align}$$
- 指数积公式只能描述末端关节对基坐标系的变换，而不能得知每个关节相对于坐标系的变换。
-
Paden-Kahan subproblem
The instantaneous spatial velocity of the end effector: $$ \begin{align} ln(g_{st}(\theta))&= \hat{\xi}1\theta_1+\dots+\hat{\xi}n\theta_n+ln(g{ab}(0)) \ \frac{\dot{g}{st}(\theta)}{g_{st}(\theta)}&=\hat{\xi}1\dot{\theta_1}+\dots+\hat{\xi}n\dot{\theta_n} \ \hat{V}^s{st}&=\dot{g}{st}(\theta)g_{st}^{-1}(\theta) \ &=\sum^n_{i=1}{(\frac{\partial{g_{st}}}{\partial{\theta_i}}\dot{\theta_i})g^{-1}{st}} \ &=\sum^n{i=1}{(\frac{\partial{g_{st}}}{\partial{\theta_i}}g^{-1}{st})\dot{\theta_i}} \ &=J^s{st}(\theta)\dot{\theta} \end{align} $$ where $J^s_{st}(\theta)=[(\frac{\partial{g_{st}}}{\partial{\theta_1}}g^{-1}{st}) \dots (\frac{\partial{g{st}}}{\partial{\theta_n}}g^{-1}_{st})]$ is the spatial manipulator Jacobian.