Robotics conventions

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

Line representations

Main article: Line coordinates

Lines are very important in robotics because:

Non-minimal vector coordinates

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

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:

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

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. 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:

  1. the z-axis is in the direction of the joint axis
  2. the x-axis is parallel to the common normal: x_n = z_n \times z_{n - 1}
    If there is no unique common normal (parallel z axes), then d (below) is a free parameter.
  3. the y-axis follows from the x- and z-axis by choosing it to be a right-handed 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.

See also

References

External links