# Geometric Description of Robots

By | September 13, 2020

Here is the summary of section 1 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

## Types of Robots

Here we aim to study some special types of robots that can describe as below:

1. Robots in which the segments are connected in series, as in human limb.

2. Our robots “arm” will usually have one fixed end in position, and at the other end will be the “hand” or “effector”(final segment of the robot).

3. In actual robots, this “hand” might be provided with mechanisms for grasping objects or with tools for performing some task.

4. Segments are rigid, thus; Possible motion $\to$ motion of joints.

5. All joints of robots lie in same plane, Therefore all motion take place in one plane.

## Parts of Robots

1. Segments

2. Planar revolute joints; permit rotations of one segment relative to another.‎

‎‎‎3. Prismatic joints; permit one segment of a robot to move by sliding or translation along an axis.

## Joint Settings

‎‎ Setting of a revolute joint: ‎The ‎totality ‎of ‎settings ‎of ‎such a‎ ‎joint ‎can ‎be ‎parametrized ‎by a‎ Circle‎‎ ‎$S^1 ‎$ ‎or ‎by the ‎interval‎ ‎$[0, 2 \pi ] ‎$ ‎with ‎the ‎endpoints ‎identified. ‎Also ‎in ‎constraint‎ ‎ ‎$\to ‎$ ‎subset ‎of‎ ‎ ‎$S^1 ‎$ .‎‎

‎Setting of a prismatic joint:‎ The settings of such a joint can be parametrized by a finite interval of real numbers, or the total length of the segment, also can be specified by giving the distance the joint is extended.‎‎

## What Is Our Purpose of This Study?

1. Describe and specify the position and orientation of the “hand”.‎‎‎

2. To illustrate how affine varieties can be used to describe the geometry of robots.

## Joint Space of A Planar Robot

‎‎If a‎ ‎planar ‎robot ‎has‎ ‎$r ‎$ ‎revolute ‎joints ‎and‎ ‎ ‎$p ‎$ ‎prismatic ‎joints, ‎then ‎the ‎possible ‎settings ‎of ‎the ‎whole ‎collection ‎of ‎it‎s joints can be parametrized by the cartesian product.‎‎

‎ ‎$‎\mathcal{‎J} = S^1 \times \ldots \times S^1 \times I_1 \times \ldots \times I_p ‎$

‎Where there is one ‎ ‎$S^1 ‎$ ‎factor ‎for ‎each ‎revolute ‎joint, ‎and‎ ‎ ‎$I_j ‎$ ‎gives ‎the ‎settings ‎of ‎the‎ ‎ ‎$j-th ‎$ ‎prismatic ‎joint. ‎We ‎call‎ ‎ ‎$‎\mathcal{J}‎ ‎$ ‎the‎ ‎joint space ‎of ‎the ‎robot.‎‎‎‎‎

## Configuration Space of The Robot’s Hand

‎By ‎fixing a‎ ‎cartesian ‎coordinate ‎system ‎in ‎the ‎plane, ‎the ‎possible ‎positions ‎of ‎the ‎”hand” ‎can ‎be ‎represented by‎ ‎the ‎points‎ ‎$(a,b) ‎$ ‎of a‎ ‎region‎ ‎ ‎$U \subseteq ‎\mathbb{R}^2‎ ‎$ . ‎‎

Also ‎the ‎orientation ‎of ‎the ‎‎”hand” ‎can ‎be ‎represented ‎by ‎giving a‎ ‎unit ‎vector ‎aligned ‎with ‎some ‎specific ‎feature ‎of ‎the ‎hand. ‎Thus‎, the possible hand orientations are parametrized by vector ‎ ‎$u ‎$ ‎in‎ ‎ ‎$V = S^1 ‎$ .‎‎‎

Now ‎we ‎call‎ ‎ ‎$‎\mathcal{C} = U ‎\times V‎ ‎$ ‎ ‎the ‎configuration space ‎or‎ ‎operational space ‎of ‎th‎e robot’s hand. ‎‎‎‎

## An Important Function

‎There ‎is a‎ ‎function ‎or ‎map ‎from ‎joint ‎space ‎of ‎the ‎robot ‎to ‎its ‎configuration ‎space:‎

‎ ‎$f: ‎\mathcal{J} \to ‎\mathcal{C}‎‎ ‎$

This function encodes how the different possible joint settings yield different hand configurations.‎‎

Question: ‎If‎ ‎ ‎$f ‎$ ‎be a‎ ‎map ‎(a continuous function), ‎what ‎are ‎the ‎topologies ‎of ‎the ‎spaces‎ ‎ ‎$‎\mathcal{J}‎ ‎$ ‎and‎ ‎ ‎$‎\mathcal{C}‎ ‎$ ?‎

‎‎ How if we change their topologies?