Solution:

Consider the case where we need to perform the manipulation of the parts where they need to be approached from above i.e. a part may be required to pick up from a horizontal surface as we can see in the figure and also the part need to be placed on the horizontal surface.

For such operations of task manipulation, we consider either the SCARA robot or 4-axis horizontal or jointed robot.

In this, the approach vector r3 is a constraint to be along a vertical line within the same plane such as:

$ r^3=-i^3\\ $

In the 4-axis SCARA robot: the first 3-axes are major axes for the position of the tooltip.

4th-axis: minor axis is for orienting the tool by changing the sliding.

When we substitute the above equation in,

$ w \triangleq\left[\begin{array}{c} w^1 \\ \hdashline w^2 \end{array}\right] \triangleq\left[\begin{array}{c} p \\ \hdashline\left[\exp \left(q_n / \pi\right)\right] r^3 \end{array}\right]\\ $

we obtain the following:

$ w=\left[p_1, p_2, p_3, 0,0,-\exp \frac{q_4}{\pi}\right]^T\\ $

We obtain 4 degrees of freedom in which 2 components of the TCV are always zero.