# Inverse Kinematic Problem

By | September 13, 2020

Here is the summary of section 3 of Chapter 6; Robotics and Automatic Geometric Theorem Proving of the book {Ideals, Varieties, and Algorithms} By‎ David A. Cox , John Little and Donal O’Shea

The ‎problem ‎is:‎‎‎

Given‎ ‎$c \in ‎\mathcal{C}‎$ , ‎can ‎we ‎determine ‎one ‎or ‎all ‎the‎ ‎ $j \in ‎\mathcal{J}‎$ ‎such ‎that‎ ‎ $f(j) = c$ ?‎‎‎

Indee‎d ‎‎we wish to determine $j$ ‎whether it is possible to place the hand of the robot at that point with that orientation. And if it is possible, we wish to find all combinations of joint settings that will accomplish this.‎

In ‎other ‎words, ‎we ‎want ‎to ‎determine ‎the ‎‎image ‎of ‎the ‎mapping‎ $f: \mathcal{J} \to \mathcal{C}$ for this robot; for each $c$ ‎in ‎the ‎image ‎of‎ $f$ ‎, we want to determine the ‎‎inverse image$f^{-1}(c)$ .‎