Line representations in robotics

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

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 alfa
• 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

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.

References

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