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. 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 $K_{n-1}$ and $K_n$ in their zero position and the corresponding common normal represented by a dashed red line. To describe the transformation of $K_n$ with respect to $K_{n-1}$, the 4 Denavit-Hartenberg parameters $\theta_n$, $d_n$, $l_n$ and $\alpha_n$ are used. These parameters describe the static transformation within link $L_n$, but as well include the dynamic influence of the joint parameter of $J_n$, that could change over time. The figure illustrates the parameters, that are defined as follows:

$\theta_n$	The angle $\theta_n$ is defined as the angle about the $z_{n-1}$-axis to align $x_{n-1}$ with the new $x_{n}$-axis. In case of a revolute joint The Denavit-Hartenberg parameters should include the dynamic influence of the joints. So as a revolute joint $J_n$ located in the coordinate frame $K_{n-1}$ allows for a rotation about the $z_{n-1}$-axis, the joint parameter (here $\theta_J$) has to be included in the Denavit-Hartenberg parameter $\theta_n$. The figure on the right shows a link $L_n$ in dark grey with the two coordinate frames $K_{n-1}$ at its beginning and $K_{n}$ at the distal joint. The coordinate frames are shown in their zero position. The angle $\theta_L$ describes the rotation about the $z_{n-1}$-axis to align the two $x$-axes. So in case of a revolute joint, there is not only the static rotation by $\theta_L$, but as well the dynamic rotation angle $\theta_J$ caused by the joint itself. Thus the parameter $\theta_n$ is the sum of the static and the dynamic rotation: $\theta_n = \theta_L + \theta_J$ Hint: In the Robotics course at the University of Paderborn, cases with a static rotation $\theta_L$ for a revolute joint are usually avoided. So for revolute joints, there is only the actual joint parameter as $\theta_n$ and the $x$-axes are aligned in their zero position: $\theta_n = \theta_J$
$d_n$	$d_n$ 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. In case of a prismatic joint If the joint $J_n$ is a prismatic joint, its degree of freedom is the translation along the $z_{n-1}$-axis. In these cases, similar to $\theta_n$ for a revolute joint,$d_n$ consists of a static and of a dynamic offset. Like can be seen in the figure on the right, the static offset $d_L$ is based on the geometric structure of the link and also valid in the zero position. The dynamic offset $d_J$ corresponds to the joint parameter and so to the displacement of joint $J_n$ and is zero in the manipulator's zero position. The displacement is illustrated in dark grey in the right part of the figure. $d_n=d_L+d_J$ Hint: In the Robotics course at the University of Paderborn, the parameter $d_n$ is usually just defined as a single value corresponding to the complete offset along the $z_{n-1}$ axis. In the figure on the right, this value is indicated as $k$. So no distinction between the static and the dynamic part is made for reasons of simplification: $d_n = k$. If there is actually a static offset, then this certain offset is just assumed as the lower limit for $k$.
$l_n$	The parameter $l_n$ corresponds to the translation along the new $x_n$-axis. The translation distance is equivalent to the length of the common normal. It has to be kept in mind, that $x_n$ and the common normal can be antiparallel and that, in such cases, a positive translation is directed in the negative direction of the common normal. If the related joint $J_n$ is a revolute joint, $l_n$ can also be regarded as the radius of the rotation about the $z_{n-1}$-axis
$\alpha_n$	The angle $\alpha_n$ corresponds to the angle about the new $x_n$-axis, which is collinear to the common normal, to align the $z_{n-1}$-axis with the new $z_{n}$-axis. So the rotation direction for positive angles depends on the direction of $x_n$.

Placement in the context of transformations

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. $\theta_n$, $d_n$, $l_n$ and $\alpha_n$ 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 $\alpha_n$ is applied followed by a translation along the same axis by $l_n$. Then the coordinate frame is rotated about the $z_{n-1}$-axis by $\theta_n$. Finally a translation along the $z_{n-1}$-axis by $d_n$ 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 videos at the end of this page explain 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 $\theta_n$ $d_n$ $l_n$ $\alpha_n$ 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 $\theta_n$ $d_n$ $l_n$ $\alpha_n$ 1 $\phi_1$ $k_1$ $-k_2$ $-90^\circ$ 2 $\phi_2$ $k_3$ $0$ $0^\circ$ 3 $0^\circ$ $k_4$ $0$ $-90^\circ$ 4 $-90^\circ+\phi_4$ $k_5$ $k_6$ $180^\circ$ 5 $\phi_5$ $0$ $k_7$ $0^\circ$ Considerable aspects of this manipulator are: For the revolute joints $J_1$, $J_2$ and $J_5$ in zero position, there is no rotation around the $z$-axes necessary to align the $x$-axes. Thus $\theta_L$ is zero in these cases and so the $\theta_n$ parameters just contain the corresponding angles $\phi_n$. At the coordinate frame $K_4$, the $x$-axis changes its direction and is no longer collinear with the previous one. A rotation by $-90^\circ$ about $z_3$ is necessary to align $x_4$ with $x_3$. As joint $J_4$ is a revolute joint with joint parameter $\phi_4$, the corresponding $\theta_4$ results in $\theta_4=-90^\circ+\phi_4$. $d_n=-k_2$ as the translation is applied in negative direction along the $x_1$-axis. $d_3=k_4$ is the only dynamic offset along the $z_{2}$-axis as the corresponding joint $J_3$ is a prismatic joint. Like explained above, the whole offset along $z_{n-1}$, here indicated with $k_4$, is used as $d_n$-parameter in case of a prismatic joint.

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