典型2R机械臂结构分析 2R-manipulator Geometric Modeling

2R-manipulator Geometric Modeling

$$({\theta }_1,{\theta }_2)\to (x,y)$$

$$\overrightarrow{OP}=\overrightarrow{OA}+\overrightarrow{AP}$$

$$\overrightarrow{OP}=l_1\overrightarrow{u}+l_2\overrightarrow{v}$$
Here,we know $\vec{u}=\left[\begin{array}{c}cos{\theta }_1\\ sin{\theta }_1\end{array}\right]$,and $\overrightarrow{v_1}=\left[\begin{array}{c}cos({\theta }_1+{\theta }_2)\\ sin({\theta }_1+{\theta }_2)\end{array}\right]$，
For the direct geometric model,we have
$$P_x=l_{1}cos{\theta }_1+l_{2}cos({\theta }_1+{\theta }_2)$$

$$P_y=l_{1}sin{\theta }_1+l_{2}sin({\theta }_1+{\theta }_2)$$

$$(x,y)\to ({\theta }_1,{\theta }_2)$$

As we would like to consider about the inverse geometric model,
$$x^2=l_1^{2}co{s}^2{\theta }_1+2l_1{l}_{2}cos{\theta }_{1}cos({\theta }_1+{\theta }_2)+l_2^{2}co{s}^2({\theta }_1+{\theta }_2)$$
$$y^2=l_1^{2}si{n}^2{\theta }_1+2l_1{l}_{2}sin{\theta }_{1}sin({\theta }_1+{\theta }_2)+l_2^{2}si{n}^2({\theta }_1+{\theta }_2)$$
We add these two equations,we know that
$$x^2+y^2=l_1^2+l_2^2+2l_1{l}_{2}cos{\theta }_2$$
So we know that
$$cos{\theta }_2=\frac{?l_1^2?l_2^2+x^2+y^2}{2l_1{l}_2}$$
$$equation(R):{\theta }_2=+/?arccos(\frac{?l_1^2?l_2^2+x^2+y^2}{2l_1{l}_2})$$

QUESTION 1:How about ${\theta }_1(x.y)$?For inverse geometric,if we know the position of the end-effector$(x,y)$,we could get 2 solution which could be observed in equation R.We have positive and negative ${\theta }_2$,and each solution has its corresponding value of ${\theta }_1$.So why there is no equation for ${\theta }_1(x.y)$. I still feel confused with it.

Feedback:knowing ${\theta }_2$, you end up with a linear system of equations with two equations and two unknown $(cos({\theta }_1)$ and $sin(\theta 1))$

Then we would like to calculate the Kinematic Jacobian Matrix.

$$\frac{\mathrm{d}P}{\mathrm{d}t}=\frac{d(\overrightarrow{OA}+\overrightarrow{AP})}{dt}=l_1\frac{\mathrm{d}\overrightarrow{u}}{\mathrm{d}t}+l_2\frac{\mathrm{d}\overrightarrow{v}}{\mathrm{d}t}=l_1\mathbf{E}\mathbf{u}\dot{{\theta }_1}+l_2\mathbf{E}\mathbf{v}(\dot{{\theta }_1}+\dot{{\theta }_2})$$
Where $$E=\left[\begin{array}{cc}0& ?1\\ 1& 0\end{array}\right]$$Then we get
$$\dot{P}=(l_1\mathbf{E}\mathbf{u}+l_2\mathbf{E}\mathbf{v})\dot{{\theta }_1}+l_2\mathbf{E}\mathbf{v}\dot{{\theta }_2}$$
We use this kind of way(using vector).
$$\dot{P}=\mathbf{E}[{\mathbf{R}}_1{\mathbf{R}}_2]\left[\begin{array}{c}\dot{{\theta }_1}\\ \dot{{\theta }_2}\end{array}\right]$$

When we talk about singularity it always equal to calculate $||J||$ or $det(J)=0$.
$$\mathbf{E}[{\mathbf{R}}_1{\mathbf{R}}_2]=\left[\begin{array}{cc}0& ?1\\ 1& 0\end{array}\right]\left[\begin{array}{cc}{l}_{1}cos{\theta }_1+l_{2}cos({\theta }_1+{\theta }_2)& l_{2}cos({\theta }_1+{\theta }_2)\\ l_{1}sin{\theta }_1+l_{2}sin({\theta }_1+{\theta }_2)& l_{2}sin({\theta }_1+{\theta }_2)\end{array}\right]$$
We have ${\theta }_2=0$ to make the $rank<2$
QUESTION2:When passing close to this singularity(maybe a type of Elbow singularity),how can we analyze what will happen to the robot?

Feedback:the kinematic Jacobian matrix gives you the motions that are not possible

We have ${\theta }_2=\pi$ to make the $rank<2$

I have found a video “Ｍeca500”,it introdruces３tpyes of singularities(Note,the last one is about the situation which is passing close to shoulder singularity)

wrist singularity	Elbow singularity	Shoulder singularity