Difference between revisions of "Denavit-Hartenberg parameters"
Line 10: | Line 10: | ||
[[File:dh-param-theta.png|right|200px]] The angle <math>\theta_n</math> is defined as the angle about the <math>z_{n-1}</math>-axis to align <math>x_{n-1}</math> with the new <math>x_{n}</math>-axis. | [[File:dh-param-theta.png|right|200px]] The angle <math>\theta_n</math> is defined as the angle about the <math>z_{n-1}</math>-axis to align <math>x_{n-1}</math> with the new <math>x_{n}</math>-axis. | ||
− | + | The joint <math>J_n</math>, that is located in the coordinate frame <math>K_{n-1}</math>, could be a revolute joint. Thus in such cases, there is not only the fixed rotation necessary to align <math>x_{n-1}</math> and <math>x_{n}</math> in their zero position, but a dynamic rotation caused by the joint itself. This is illustrated in the figure on the right. The dark grey part is the link <math>L_n</math>. The coordinate frame <math>K_{n-1}</math> is attached to the joint at its beginning and the next frame <math>K_n</math< is located at the distal joint. Regarding the zero position, there could be a rotation within the link around the <math>z_{n-1}</math>-axis to align the two <math>x</math>-axes. This angle is indicated as <math>\theta_L</math> in the figure. | |
|- | |- | ||
|style="background-color:#e8e8e8;"|<math>d_n</math> | |style="background-color:#e8e8e8;"|<math>d_n</math> |
Revision as of 11:47, 16 November 2015
← Back: Assigning coordinate frames | Overview: Denavit-Hartenberg Convention | Next: A-matrices → |
When the coordinate frames are assigned to a manipulator, the transformation between each two consecutive frames has to be described. As before for the assignment of the coordinate frames, the manipulator has to be in its zero position as well for the determination of the parameters. The figure on the right shows the two coordinate frames and in their zero position and the corresponding common normal represented by a dashed red line. To describe the transformation of with respect to , the 4 Denavit-Hartenberg parameters , , and are used. The figure illustrates the parameters, that are defined as follows:
The angle is defined as the angle about the -axis to align with the new -axis.
The joint , that is located in the coordinate frame , could be a revolute joint. Thus in such cases, there is not only the fixed rotation necessary to align and in their zero position, but a dynamic rotation caused by the joint itself. This is illustrated in the figure on the right. The dark grey part is the link . The coordinate frame is attached to the joint at its beginning and the next frame -axis to align the two -axes. This angle is indicated as in the figure. | |
is the offset or translation, respectively, along the -axis from the origin of to the intersection with the common normal. | |
The parameter corresponds to the length of the common normal which is equivalent to the translation along it. If the related joint is a revolute joint, can also be regarded as the radius of the rotation about the -axis | |
The angle corresponds to the angle about the common normal to align the -axis with the new -axis |
The 4 parameters can rather be determined by just regarding the two coordinate frames, their axes and the common normal like visualized above. To completely understand the parameters and their meaning, the figure below illustrates what the parameters actually describe. , , and define 4 transformations that are applied consecutively to transform the coordinate frame to . First a rotation about the -axis by is applied followed by a translation along it by . Then the coordinate frame is rotated about the -axis by . Finally a translation along the -axis leads to the next coordinate frame . Some further aspects about the meaning and the use of the 4 parameters are described in the following article about the A-matrices.
The video at the end of this page explains the assignment of the coordinate frames and the determination of the 4 parameters very vividly and comprehensibly.
|
Example: Determination of the Denavit-Hartenberg parameters
The table below contains the Denavit-Hartenberg parameters for the manipulator shown in the figure on the right. For further information about the already assigned coordinate frames, have a look on the examples of the previous articles. The necessary lengths of certain parts of the manipulator are indicated by the variables to .
|
Multimedial educational material
https://www.youtube.com/watch?v=qZB3_gKBwf8 Video: Assignment of coordinate frames and determination of the parameters (in German) |