Solution:

To solve the inverse kinematics problem, the input is the desired tool configuration. In order to obtain the tool, roll angle from a scaled approach we should use an invertible function of the roll angle qn to scale the r3 length.

When qn is bounded then we obtain the following positive, invertible, exponential scaling function.

$ f\left(q_n\right) \triangleq \exp \frac{q_n}{\pi}\\ $

Tool Configuration vector(TCV):

Assume that the p and R are the position and orientation of the tool with respect to the Base frame.

qn = toll roll angle

TCV = vector in R6

$ w \Delta\left[\begin{array}{c} w^{\prime} \\ \cdots-\cdot \\ 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]\\ $

$\mathrm{W} 1=\mathrm{p}=$ tool tip position

W2 = last 3 components present the tool orientation

qn can be determined from the TCV i.e. W

Tool Roll:

The following equation provides that the tool roll angle can be obtained from the TCV w

$ q_n=\pi \ln \left(w_4^2+w_3^3+w^2\right)^{1 / 2}\\ $

The TCS parameters i.e. p and R are always associated with a vector ‘w’ which is a a subset of R6 .

This R6 is called TCS and is a 6D frame.

The tool or the gripper or the end–effector is lying in its own space or rotating in its own space. That space is called TCS. Joint Space or Joint Coordinate Space.

The n-dimensional space in which all the n no of joints is situated is called space or the vector space and is denoted by Rn.