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There are a lot of conventions used in the Robotics research field. This article summarises these conventions. =Line representations= Lines are very important in robotics because:
NON-MINIMAL VECTOR COORDINATES A line is completely defined by the ordered set of two vectors:
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:
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
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 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 DH representation has problems to represent parallel lines, since for parallel lines
These two effects are examples of coordinate singularities. This problem can be solved in two ways:
Hayati-Roberts line coordinates The Hayati-Roberts line representation, denoted , 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 or 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. DENAVIT-HARTENBERG LINK FRAME CONVENTION (DH) HAYATI-ROBERTS LINK FRAME CONVENTION (HR) =See Also=
=References=
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