Line representations in robotics

Line representations in robotics are used for the following:

When using such line it is needed to have conventions for the representations so they are clearly defined. This article discusses several of these methods.

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

Each point ${\displaystyle x}$ on the line is given a parameter value ${\displaystyle t}$ that satisfies: ${\displaystyle x=p+td}$. The parameter t is unique once ${\displaystyle p}$ and ${\displaystyle d}$ are chosen.
The representation ${\displaystyle L(p,d)}$ is not minimal, because it uses six parameters for only four degrees of freedom.
The following two constraints apply:

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 ${\displaystyle L_{pl}(d,m)}$. Both ${\displaystyle d}$ and ${\displaystyle m}$ are free vectors: ${\displaystyle d}$ represents the direction of the line and ${\displaystyle m}$ is the moment of ${\displaystyle d}$ about the chosen reference origin. ${\displaystyle m=p\times d}$ (${\displaystyle m}$ is independent of which point ${\displaystyle p}$ 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

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).

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: