convert toolframe coordinate to world frame coordinates?

Question

I am not sure how i should explain this, I am looking for a way to plot the trajectory an robot arm. An object is seen from the toolFrame frame, but how do I plot the position of each joint, such that the frame they uses are the same.

One way would be to use the world frame as reference, but how would i plot the position of the object related to the world frame?

Answer 1

Assuming you have solved the inverse kinematics (IK) problem already...

I suppose you have a transformation matrix for each joint, build up from one line in the DH table (if you used DH to describe the robot). Form the IK you have obtained

$Q = [q_1, q_2, q_3, q_4, q_5, q_6]$

Having all Q values you can now write:

$H_{0,1} = A_1(q_1)$

$\Rightarrow$ $H_{0,1}(1:3, 4)$ will give you the $x$, $y$ and $z$ Coordinates of joint 1

$H_{0,2} = H_{0,1}*A_2(q_2)$

$\Rightarrow$ $H_{0,2}(1:3, 4)$ will give you the $x$, $y$ and $z$ Coordinates of joint 2

$H_{0,3} = H_{0,2}*A_3(q_3)$

$\Rightarrow$ $H_{0,3}(1:3, 4)$ will give you the $x$, $y$ and $z$ Coordinates of joint 3

$H_{0,4} = H_{0,3}*A_4(q_4)$

$\Rightarrow$ $H_{0,4}(1:3, 4)$ will give you the $x$, $y$ and $z$ Coordinates of joint 4

$H_{0,5} = H_{0,4}*A_5(q_5)$

$\Rightarrow$ $H_{0,5}(1:3, 4)$ will give you the $x$, $y$ and $z$ Coordinates of joint 5

$H_{0,6} = H_{0,5}*A_6(q_6)$

$\Rightarrow$ $H_{0,6}(1:3, 4)$ will give you the $x$, $y$ and $z$ Coordinates of joint 6

$H_{0,tool} = H_{0,6}*A_{tool}$

where

• $A_i$ is the DH-transformation matrix associated with joint $i$.
• the index $_0$ represents the world frame
• the intex $_{tool}$ represents the tool frame
• $H_{i,j}$ represent the transformation matrix from frame $i$ to frame $j$