# Robotics conventions

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

## Line representations

{{#invoke:main|main}} Lines are very important in robotics because:

• They model joint axes: a revolute joint makes any connected rigid body rotate about the line of its axis; a prismatic joint 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 ${\displaystyle L(p,d)}$ is completely defined by the ordered set of two vectors:

Each point ${\displaystyle x}$ on the line is given a parameter value ${\displaystyle t}$ that satisfies: ${\displaystyle x=p+td}$. The parameter t is unique once ${\displaystyle p}$ and ${\displaystyle d}$ are chosen.
The representation ${\displaystyle L(p,d)}$ 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 ${\displaystyle L_{pl}(d,m)}$. Both ${\displaystyle d}$ and ${\displaystyle m}$ are free vectors: ${\displaystyle d}$ represents the direction of the line and ${\displaystyle m}$ is the moment of ${\displaystyle d}$ about the chosen reference origin.${\displaystyle m=p\times d}$ (${\displaystyle m}$ is independent of which point ${\displaystyle p}$ 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

• 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

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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 ${\displaystyle z}$-axis is in the direction of the joint axis
2. the ${\displaystyle x}$-axis is parallel to the common normal: ${\displaystyle x_{n}=z_{n}\times z_{n-1}}$
If there is no unique common normal (parallel ${\displaystyle z}$ axes), then ${\displaystyle d}$ (below) is a free parameter.
3. the ${\displaystyle y}$-axis follows from the ${\displaystyle x}$- and ${\displaystyle z}$-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 ${\displaystyle L_{hr}(e_{x},e_{y},l_{x},l_{y})}$, 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 ${\displaystyle X}$ or ${\displaystyle Y}$ axis of the world frame.

## References

• Giovanni Legnani, Federico Casolo, Paolo Righettini and Bruno Zappa A homogeneous matrix approach to 3D kinematics and dynamics — I. Theory Mechanism and Machine Theory, Volume 31, Issue 5, July 1996, Pages 573–587
• Giovanni Legnani, Federico Casalo, Paolo Righettini and Bruno Zappa A homogeneous matrix approach to 3D kinematics and dynamics—II. Applications to chains of rigid bodies and serial manipulators Mechanism and Machine Theory, Volume 31, Issue 5, July 1996, Pages 589–605
• A. Bottema and B. Roth. Theoretical Kinematics. Dover Books on Engineering. Dover Publications, Inc. Mineola, NY, 1990
• A. Cayley. On a new analytical representation of curves in space. Quarterly Journal of Pure and Applied Mathematics,3:225–236,1860
• K.H. Hunt. Kinematic Geometry of Mechanisms. Oxford Science Publications, Oxford, England, 2n edition, 1990
• J. Plücker. On a new geometry of space. Philosophical Transactions of the Royal Society of London, 155:725–791, 1865
• J. Plücker. Fundamental views regarding mechanics. Philosophical Transactions of the Royal Society of London, 156:361–380, 1866
• J. Denavit and R.S. Hartenberg. A kinematic notation for lower-pair mechanisms based on matrices. Trans ASME J. Appl. Mech, 23:215–221,1955
• R.S. HartenBerg and J. Denavit Kinematic synthesis of linkages McGraw–Hill, New York, NY, 1964
• R. Bernhardt and S.L. Albright. Robot Calibration, Chapman & Hall, 1993
• S.A. Hayati and M. Mirmirani. Improving the absolute positioning accuracy of robot manipulators. J. Robotic Systems, 2(4):397–441, 1985
• K.S. Roberts. A new representation for a line. In Proceedings of the Conference on Computer Vision and Pattern Recognition, pages 635–640, Ann Arbor, MI, 1988