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. Thus, the resulting velocities $\dot{q}$ complies with the law and at the same time will be the closest to $\dot{q_0}$ in the least-squares sense.
Further, the equation (3.9) is clearly made up of two contributions:
- $J^+v$ that is the primary task as defined by $v$.
- $\left( I -J^+J \right)\dot{q_0}$ that is the secondary task as defined by $\dot{q_0}$.
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 offered by the redundancy of the manipulator at hand 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 relation:
$$
\dot{q_0} = k_0 \frac{\partial w(q)}{\partial q},
$$
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 set:
$$
w(q) = -\frac{1}{2n}\sum^n_{i=1}\left( \frac{q_i-\bar{q_i}}{q_{iM}-q_{im}} \right)^2,
$$
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, for example, 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 to it nonnull angular velocities.
