Difference between revisions of "Denavit-Hartenberg parameters"

Revision as of 18:17, 16 November 2015

Overview: Denavit-Hartenberg Convention

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 length of the common normal which is equivalent to the translation along it.

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 common normal to align the $z_{n-1}$ -axis with the new $z_{n}$ -axis

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. $\theta$ , $d$ , $l$ 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	$\theta_n$	$d_n$	$l_n$	$\alpha_n$
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 $k_7$ .

Considerable aspects of this manipulator are:

$d_n=-k_2$ 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.
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 $\psi_4$ , the corresponding parameter results in $\theta_4=-90^\circ+\psi_4$ .

T	$\theta_n$	$d_n$	$l_n$	$\alpha_n$
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	$\theta_5$	$0$	$k_7$	$0^\circ$

Multimedial educational material

https://www.youtube.com/watch?v=qZB3_gKBwf8 Video: Assignment of coordinate frames and determination of the parameters (in German)

@@ Line 85: / Line 85: @@
 Considerable aspects of this manipulator are:
-*  <math>d_n</math> is <math>-k_2</math> as the translation is applied in negative direction along the <math>x_1</math>-axis.
+* <math>d_n=-k_2</math> as the translation is applied in negative direction along the <math>x_1</math>-axis.
+* <math>d_3=k_4</math> is the only dynamic offset along the <math>z_{2}</math>-axis as the corresponding joint <math>J_3</math> is a prismatic joint. Like explained above, the whole offset along <math>z_{n-1}</math>, here indicated with <math>k_4</math>, is used as <math>d_n</math>-parameter in case of a prismatic joint.
+* At the coordinate frame <math>K_4</math>, the <math>x</math>-axis changes its direction and is no longer collinear with the previous one. A rotation by <math>-90^\circ</math> about <math>z_3</math> is necessary to align <math>x_4</math> with <math>x_3</math>. As joint <math>J_4</math> is a revolute joint with joint parameter <math>\psi_4</math>, the corresponding parameter results in <math>\theta_4=-90^\circ+\psi_4</math>.
 <table style="background-color: #444444">

Difference between revisions of "Denavit-Hartenberg parameters"

Revision as of 18:17, 16 November 2015

Navigation menu

Personal tools

Namespaces

Variants

Views

More

Search

Navigation

Tools