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.
is completely defined by the ordered set of two vectors: Each point
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
represents the direction of the line and
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.
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: The Hayati–Roberts line representation, denoted
, is another minimal line representation, with parameters: This representation is unique for a directed line.
The product of exponentials formula represents the kinematics of an open-chain mechanism as the product of exponentials of twists, and may be used to describe a series of revolute, prismatic, and helical joints.