If you multiply both members of the equation (3.9) by $J$, you'll get: $$ J\dot{q} = JJ^+v + \left( J -JJ^+J \right)\dot{q_0}. $$
Then, we can exploit that $JJ^+=I$, obtaining: $$ J\dot{q}=v, $$ which is the well-known forward differential kinematics law.
Until now, we didn't specify anything about $\dot{q_0}$, which indeed can be whatever vector of the same size of $\dot{q}$, meaning we can plug whatever $\dot{q_0}$ into (3.9) that the fundamental differential law will be still guaranteed.
Further, the equation (3.9) is clearly made up of two contributions:
- $J^+v$ that is the primary task.
- $\left( I -J^+J \right)\dot{q_0}$ that is the secondary task.
Because $\dot{q_0}$ can be anything, we say that the secondary task does not interfere with the primary task thanks to the projector $\left( I -J^+J \right)$.
Therefore, the flexibility given by the redundant mechanism at hand (i.e., dealing with the pseudoinverse $J^+$ and not the pure inverse $J^{-1}$ means that the manipulator is redundant) allows us to employ $\dot{q_0}$ for achieving a supplementary goal (i.e., the secondary task).
Usual secondary tasks are implemented to stay away from the joint bounds or to improve manipulability.
To this end, one can thus establish the following: $$ \dot{q_0} = k_0 \frac{\partial w(q)}{\partial q}, $$ where $w(q)$ is some sort of function of the joints $q$ that we aim to minimize/maximize through its gradient.
For example, if we want to stay away from the joint limits while converging to the primary target defined by $v$, we could do: $$ w(q) = -\frac{1}{2n}\sum^n_{i=1}\left( \frac{q_i-\bar{q_i}}{q_{iM}-q_{im}}, \right)^2 $$ with $q_{iM}$, $q_{im}$ the upper and lower bounds of joint $i$, and $\bar{q_i} = \frac{q_{iM}+q_{im}}{2}$.
Just a concluding remark about the following statement:
For robot, the null space of Jacobian is the set of joints' velocities vectors that yield zero linear and angular velocities of the end-effector
This is often the case but it's not 100% true. It very depends on the primary task, which in turn dictates the form of the Jacobian $J$.
If the primary task aims to get full control of the tip frame (i.e., linear and angular velocity), then the statement above holds outright.
However, if our primary task deals only with the position of the tip frame and not its orientation ($J$ is defined in $\mathbb{R}^{3 \times n}$), then the secondary task may impact the orientation of the end-effector by applying nonnull angular velocities.