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 $\alpha \,$
• 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.

See Also

• List of basic robotics topics

References

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