Abstract
We study the motion of a robotic arm inside a rectangular tunnel of width 2. We prove that the configuration space S of all possible positions of the robot is a CAT(0) cubical complex. Before this work, very few families of robots were known to have CAT(0) configuration spaces. This property allows us to move the arm optimally from one position to another.
Highlights
We consider a robotic arm Rn of length n moving in a rectangular tunnel of width 2 without selfintersecting
The robot consists of n links of unit length, attached sequentially, and its base is affixed to the lower left corner
Problem 1.1 Find the fastest way of moving the robotic arm Rn from one position to another
Summary
We consider a robotic arm Rn of length n moving in a rectangular tunnel of width 2 without selfintersecting. Theorem 1.2 The configuration space Sn of the pinned-down robotic arm Rn of length n in a tunnel of width 2 is a CAT(0) cubical complex. It follows from very general results of Abrams and Ghrist [1] that the configuration space Sn is a cubical complex. Theorem 1.3 Let Sn be the configuration space for the robotic arm Rn of length n moving in a rectangular tunnel of width 2. We use the PIP P (X) as a “remote control” to move the robot and navigate the space X Using this remote control, we implement an algorithm to move the robotic arm in a tunnel of width 1 (using the results of [2]) and 2 (using Proposition 1.5) optimally, solving Problem 1.1
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
