I have a script written by someone else that outputs end-effector velocities. I need to transform
these end-effector velocities to joint velocities. This requires the pseudo-inverse of the Jacobian matrix $J(q)^{\dagger}$. The equation is shown below.
$\dot{q}=J(q)^{\dagger}\dot{x}$
The problem I am now facing is how to determine this pseudo inverse.
I know that for a very basic 2 DOF, $x, y$ manipulator we have the standard forward velocity kinematics:
$\dot{x}=J(q)\dot{q}$ in which:
$\begin{bmatrix}
\dot{x} \\
\dot{y} \\
\end{bmatrix} =
\begin{bmatrix}
\frac{\partial x}{\partial q_1} & \frac{\partial x}{\partial q_2} \\
\frac{\partial y}{\partial q_1} & \frac{\partial y}{\partial q_2} \\\end{bmatrix}
\begin{bmatrix}
\dot{q_1} \\
\dot{q_2} \\\end{bmatrix}$
We can rewrite this expression to obtain the joint velocities:
$\dot{q}=J(q)^{\dagger}\dot{x}$ in which:
$\begin{bmatrix}
\dot{q_1} \\
\dot{q_2} \\\end{bmatrix} =
\begin{bmatrix}
?\end{bmatrix}
\begin{bmatrix}
\dot{x} \\
\dot{y} \\\end{bmatrix}$
I am wondering what should be put in the place of the question mark?
Best Answer
I have created a powerpoint solving my question. Instead of re-entering all the matrices here, I simply paste images of the powerpoint slides. I hope this helps someone in the future: