# Robotics conventions

There are many conventions used in the robotics research field. This article summarises these conventions.

## Line representations

{{#invoke:main|main}} Lines are very important in robotics because:

• They model joint axes: a revolute joint makes any connected rigid body rotate about the line of its axis; a prismatic joint makes the connected rigid body translate along its axis line.
• They model edges of the polyhedral objects used in many task planners or sensor processing modules.
• They are needed for shortest distance calculation between robots and obstacles

## Non-minimal vector coordinates

A line $L(p,d)$ is completely defined by the ordered set of two vectors:

### Plücker coordinates

Arthur Cayley and Julius Plücker introduced an alternative representation using two free vectors. This representation was finally named after Plücker.
The Plücker representation is denoted by $L_{pl}(d,m)$ . Both $d$ and $m$ are free vectors: $d$ represents the direction of the line and $m$ is the moment of $d$ about the chosen reference origin.$m=p\times d$ ($m$ is independent of which point $p$ on the line is chosen!)
The advantage of the Plücker coordinates is that they are homogeneous.
A line in Plücker coordinates has still four out of six independent parameters, so it is not a minimal representation. The two constraints on the six Plücker coordinates are

• the homogeneity constraint
• the orthogonality constraint

## Minimal line representation

A line representation is minimal if it uses four parameters, which is the minimum needed to represent all possible lines in the Euclidean Space (E³).

### Denavit–Hartenberg line coordinates

{{#invoke:main|main}}

Jaques Denavit and Richard S. Hartenberg presented the first minimal representation for a line which is now widely used. The common normal between two lines was the main geometric concept that allowed Denavit and Hartenberg to find a minimal representation. Engineers use the Denavit–Hartenberg convention(D–H) to help them describe the positions of links and joints unambiguously. Every link gets its own coordinate system. There are a few rules to consider in choosing the coordinate system:

Once the coordinate frames are determined, inter-link transformations are uniquely described by the following four parameters:

### Hayati–Roberts line coordinates

The Hayati–Roberts line representation, denoted $L_{hr}(e_{x},e_{y},l_{x},l_{y})$ , is another minimal line representation, with parameters:

This representation is unique for a directed line. The coordinate singularities are different from the DH singularities: it has singularities if the line becomes parallel to either the $X$ or $Y$ axis of the world frame.