There are a lot of conventions used in the Robotics research field. This article summarises these conventions.

Line representations

Lines are very important in robotics because:

The model joint axes: a revolute joint makes any connected rigid 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:

a point vector $p$ , indicating the position of an arbitrary point on $L$
one free direction vector $d$ , giving the line a direction as well as a sense.

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

The direction vector $d$ can be chosen to be a unit vector
the point vector $p$ can be chosen to be the point on the line that is nearest the origin. So $p$ is orthogonal to $d$

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. $m=pxd$ ( $m$ is independent of which point $p$ on the line is chosen!)
The advantage of the Plücker coordinates is that they are homogenous.
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

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. The line L must first be given a direction, and is then uniquely described by the following four parameters:

The distance d
The azimuth $\alpha \,$
The twist $\theta \,$
The height h

The literature contains alternative formulations, differing mainly in the conventions for signs and reference axes. Conceptually, all these formulations are equivalent, and they represent the line L by two translational and two rotational parameters.
Note that a set of four DH parameters not only represents a line, but also the pose of a frame, that has its Z axis on the given line and its X axis along the common normal.
Since only four parameters are used, the frames that can be represented this way satisfy two constraints

the frame's X-axis intersects the Z-axis of the world frame
the frame's X-axis is parallel to the XY-plane of the world frame.

The DH representation has problems to represent parallel lines, since for parallel lines

the common normal is not uniquely defined
the parameters change discontinuously when the line moves continuously through a configuration in which it is parallel to the Z-axis of the world frame

These two effects are examples of coordinate singularities. This problem can be solved in two ways:

Using more than one coordinate patch
using more than four parameters for a line

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:

$e_{x}$ and $e_{x}$ are the $X$ and $Y$ components of a unit direction vector $e$ on the line. This requirement eliminates the need for a $Z$ component, since $e_{z}=(1-e_{x}^{2}-e_{y}^{2})^{\frac {1}{2}}$
$l_{x}$ and $l_{y}$ are the coordinates of the intersection point of the line with the plane through the origin of the world reference frame, and normal to the line. The reference frame on this normal plane has the same origin as the world reference frame, and its $X$ and $Y$ frame axes are images of the world frame's $X$ and $Y$ axes throug parallel projection along the line.

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

Link Frame Conventions

Coordinate representations of robotic devices have to allow to represent the relative pose and velocity of two neighbouring links, as a function of the position and velocity of the joint connecting both links.

Denavit-Hartenberg link frame convention (DH)

Hayati-Roberts link frame convention (HR)

References

Bruyninckx, Herman, De Schutter, Joris. Introduction to intelligent robotics.

Line representations in robotics