Denavit-Hartenberg parameters

Overview: Denavit-Hartenberg Convention

When the coordinate frames are assigned to a manipulator, the transformation between each two consecutive frames has to be described. Therefor the 4 Denavit-Hartenberg parameters $\theta$ , $d$ , $l$ and $\alpha$ are used. 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 two coordinate frames $K_{n-1}$ and $K_n$ and the corresponding common normal represented by a dashed red line. The figure illustrates the Denavit-Hartenberg parameters, that are defined as follows:

$\theta$	The angle $\theta$ is defined as the angle about the $z_{n-1}$ -axis to align $x_{n-1}$ with the new $x_{n}$ -axis.
$d$	$d$ is the offset or translation, respectively, along the $z_{n-1}$ -axis from the origin of $K_{n-1}$ to the intersection with the common normal.
$l$	The parameter $l$ corresponds to the length of the common normal which is equivalent to the translation along it. If the related joint $J_n$ is a revolute joint, $l$ can also be regarded as the radius of the rotation about the $z_{n-1}$ -axis
$\alpha$	The angle $\alpha$ corresponds to the angle about the common normal to align the $z_{n-1}$ -axis with the new $z_{n}$ -axis

Special case: It can occur that there is an offset ...

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. $\theta$ , $d$ , $l$ and $\alpha$ define 4 transformations that are applied consecutively to transform the coordinate frame $K_{n-1}$ to $K_n$ . First a rotation about the $x_{n-1}$ -axis by $\alpha$ is applied followed by a translation along it by $l$ . Then the coordinate frame is rotated about the $z_{n-1}$ -axis by $\theta$ . Finally a translation along the $z_{n-1}$ -axis leads to the next coordinate frame $K_n$ . 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.

Hint: Notation of the Denavit-Hartenberg parameters
The Denavit-Hartenberg parameters are usually noted in a table with columns for the parameters and a row for each link or transformation, respectively.

T	$\theta$	$d$	$l$	$\alpha$
1	$\theta_1$	$d_1$	$l_1$	$\alpha_1$
...

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 $k_1$ to $k_7$ .

T	$\theta$	$d$	$l$	$\alpha$
1	$\theta_1$	$k_1$	$-k_2$	$-90^\circ$
2	$\theta_2$	$k_3$	$0$	$0^\circ$
3	$0^\circ$	$k_4$	$0$	$-90^\circ$
4	$-90^\circ+\theta_4$	$k_5$	$k_6$	$180^\circ$
5	$\theta_5$	$0$	$k_7$	$0^\circ$