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?
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: