$$
M \Delta\ddot{x} + K_d \Delta\dot{x} + h_e = h $$
where, M = Mass Matrix
K d = Damping Matrix
he = Wrench at the end effector
h = Total wrench
Now, neglecting the non- dominant terms considering we are moving the system quasi statically, we get:
$$
K_d \Delta\dot{x} + h_e = h
\implies K_d \Delta\dot{x} + (h_e - h) = 0
$$
Now multiplying Kd inverse both the sides, we get:
$$ \Delta\dot{x} + K_d^{-1} (h_e - h) =0 $$
The above equation can be written as:
$$ \dot{x_r} = \dot{x_d}+K_d^{-1} (h_e - h) $$
Now, we know that,
$$ \dot{x}= J\dot{q} $$
Multiplying J inverse in the above equation, we get:
$$ J^{-1}\dot{x_r}=J^{-1}\dot{x_d}+J^{-1}K_d^{-1}(h_e-h) \implies \dot{q_r}= \dot{q_d}+J^{-1}[K_d^{-1}(h_e-h)] $$