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

Line representations

Lines are very important in robotics because:

• They model joint axes: a revolute joint
  Revolute joint
  A revolute joint is a one degree of freedom kinematic pair used in mechanisms. Revolute joints provide single-axis rotation function used in many places such as door hinges, folding mechanisms, and other uni-axial rotation devices. -See also:* Cylindrical joint* Kinematics* Degrees of freedom *...
  
   makes any connected rigid body rotate about the line of its axis; a prismatic joint
  Prismatic joint
  A prismatic joint provides a linear sliding movement between two bodies, and is often called a slider, as in the slider-crank linkage. A prismatic joint is formed with a polygonal cross-section to resist rotation...
  
   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 is completely defined by the ordered set of two vectors:

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

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

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

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 . Both and are free vectors: represents the direction of the line and is the moment of about the chosen reference origin. ( is independent of which point 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

Common normal (robotics)

In robotics the common normal of two non-intersecting joint axes is a line perpendicular to both axes.The common normal can be used to characterize robot arm links, by using the "common normal distance" and the angle between the link axes in a plane perpendicular to 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

Coordinate system

In geometry, a coordinate system is a system which uses one or more numbers, or coordinates, to uniquely determine the position of a point or other geometric element. The order of the coordinates is significant and they are sometimes identified by their position in an ordered tuple and sometimes by...

. There are a few rules to consider in choosing the coordinate system:

1. the -axis is in the direction of the joint axis
2. the -axis is parallel to the common normal:
   If there is no unique common normal (parallel axes), then (below) is a free parameter.
3. the -axis follows from the - and -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:

• : angle about previous , from old to new
• : offset along previous to the common normal
• : length of the common normal (aka , but if using this notation, do not confuse with ). Assuming a revolute joint, this is the radius about previous .
• : angle about common normal, from old axis to new axis

Hayati–Roberts line coordinates

The Hayati–Roberts line representation, denoted , is another minimal line representation, with parameters:

• and are the and components of a unit direction vector on the line. This requirement eliminates the need for a component, since
• and 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 and frame axes are images of the world frame's and axes through 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 or axis of the world frame.

External links

• Denavit Hartenburg Convention Computational Software, Wolfram.com 'Math Source' Author: Jason Desjardins 2002