Denavit-Hartenberg parameters

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Overview: Denavit-Hartenberg Convention

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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. 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

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 they actually describe. $\theta$, $d$, $l$ and $\alpha$ define 4 transformations that are applied consecutively to transform a coordinate frame $K_{n-1}$ to $K_n$. Assume there is a coordinate frame $K_n^'$ coincident with $K_{n-1}$. This coordinate frame is first rotated about the $x$-axis by $\alpha$ and then translated along it by $l$. After that, a rotation about the $z$-axis by $\theta$ is applied. Finally a translation along the $z$