Denavit–Hartenberg parameters
Jacques Denavit and Richard Hartenberg introduced this convention in 1955 in order to standardize the coordinate frames for spatial linkages.
[1][2] Richard Paul demonstrated its value for the kinematic analysis of robotic systems in 1981.
A commonly used convention for selecting frames of reference in robotics applications is the Denavit and Hartenberg (D–H) convention which was introduced by Jacques Denavit and Richard S. Hartenberg.
The coordinate transformations along a serial robot consisting of n links form the kinematics equations of the robot: where [T ] is the transformation that characterizes the location and orientation of the end-link.
To determine the coordinate transformations [Z ] and [X ], the joints connecting the links are modeled as either hinged or sliding joints, each of which has a unique line S in space that forms the joint axis and define the relative movement of the two links.
The system of six joint axes Si and five common normal lines Ai,i+1 form the kinematic skeleton of the typical six degree-of-freedom serial robot.
This convention allows the definition of the movement of links around a common joint axis Si by the screw displacement: where θi is the rotation around and di is the sliding motion along the z-axis.
Each of these parameters could be a constant depending on the structure of the robot.
Under this convention the dimensions of each link in the serial chain are defined by the screw displacement around the common normal Ai,i+1 from the joint Si to Si+1, which is given by where αi,i+1 and ri,i+1 define the physical dimensions of the link in terms of the angle measured around and distance measured along the X axis.
In summary, the reference frames are laid out as follows: The following four transformation parameters are known as D–H parameters:[4] There is some choice in frame layout as to whether the previous x axis or the next x points along the common normal.
The latter system allows branching chains more efficiently, as multiple frames can all point away from their common ancestor, but in the alternative layout the ancestor can only point toward one successor.
We can note constraints on the relationships between the axes: It is common to separate a screw displacement into product of a pure translation along a line and a pure rotation about the line,[5][6] so that and Using this notation, each link can be described by a coordinate transformation from the concurrent coordinate system to the previous coordinate system.
In some books, the order of transformation for a pair of consecutive rotation and translation (such as
This is possible (despite the fact that in general, matrix multiplication is not commutative) since translations and rotations are concerned with the same axes
As matrix multiplication order for these pairs does not matter, the result is the same.
The Denavit and Hartenberg notation gives a standard (distal) methodology to write the kinematic equations of a manipulator.
This is especially useful for serial manipulators where a matrix is used to represent the pose (position and orientation) of one body with respect to another.
An important property of Denavit and Hartenberg matrices is that the inverse is where
Further matrices can be defined to represent velocity and acceleration of bodies.
The components of velocity and acceleration matrices are expressed in an arbitrary frame
and transform from one frame to another by the following rule For the dynamics, three further matrices are necessary to describe the inertia
represent the position of the center of mass, and the terms
momentum All the matrices are represented with the vector components in a certain frame
follows the rule The matrices described allow the writing of the dynamic equations in a concise way.
(force equal mass times acceleration) plus
Some books such as Introduction to Robotics: Mechanics and Control (3rd Edition) [7] use modified (proximal) DH parameters.
The difference between the classic (distal) DH parameters and the modified DH parameters are the locations of the coordinates system attachment to the links and the order of the performed transformations.
Compared with the classic DH parameters, the coordinates of frame
Another difference is that according to the modified convention, the transform matrix is given by the following order of operations: Thus, the matrix of the modified DH parameters becomes Note that some books (e.g.:[8]) use
Surveys of DH conventions and its differences have been published.
