I have doubts on how to find all the joint velocities that don't produce any linear velocity, so are in the Null space, when we are in a singular configuration. For example, suppose we have a 2R planar robotic arm. If we compute the Jacobian we have:
$J(q)= \begin{bmatrix}-l_1sin(q1)-l_2sin(q1+q2) & -l_2sin(q1+q2)\\ l_1cos(q1)+l_2cos(q1+q2) & l_2cos(q1+q2)\end{bmatrix} $
and if we consider the sigular configuration for which $q_2=0$ we have:
$J(q)= \begin{bmatrix}-(l_1+l_2)sin(q1) & -l_2sin(q1)\\ (l_1+l_2)cos(q1) & l_2cos(q1)\end{bmatrix} $
now i want to find the joint velocities that produce no linear velocity of the end-effector, so all the velocities that arein the Null space. I have that those are all the velocities of the type :
$N(J)= \alpha \begin{bmatrix} l_2\\ -(l_1+l_2) \end{bmatrix}$
but i don't understand how this result is obtained.
Can somebody please help me?
Thank's in advance.