Solution:

Interpolated motion is a type of motion exhibited by point-to-point robots. In many instances, the the path will not be completely specified.

Knot points such as intermediate via points and endpoints will be specified.

Interpolated motion is called for when the path that an object takes through space is important.

The trajectory planning interpolates between the knot points to produce a smooth the trajectory that can be executed using continuous-path motion control techniques.

In a general interpolation problem, the sequence of m knot points in tool configuration space is given by $\Gamma_m=\left\{w^0, \mathbf{w}^1, \ldots \ldots, \mathbf{w}^{n-1}\right\}$

The path w (t) between the m knot points should be smooth. It should have a minimum of two continuous derivatives in order to avoid the need for infinite acceleration.