Assigning coordinate frames

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To be able to determine the spatial relationship or transformation, respectively, between the links of a manipulator, local coordinate frames have to be assigned to them first. There are several rules that have to be observed when coordinate frames are to be assigned following the Denavit-Hartenberg convention. The first rule is, that the manipulator has to be moved to its zero position. So all the joints or their joint parameters, respectively, have to be set to zero. Then the coordinate frames are assigned regarding the zero position.

Following the notation, a coordinate frame is attached to each end of a link at the corresponding joint. The orientation of the coordinate frames depends on the joint and on the prior frame. The following steps describe how the coordinate frames are determined:

First coordinate frame K_0
Following the notation, the first coordinate frame is always the base or reference frame K_0. The origin of the base is coincident with the origin of the first joint and their axes are aligned. So the joint axis of the first frame is normal to the xy-plane of the base or world frame, respectively. As the world frame is usually known, the orientation of K_0 is already defined and does not have to be determined.
Origin of the frames
According to the notation, the intermediate frames K_1 to K_{k-1} are attached to the distal joints. The exact origin of frame K_n corresponds to the intersection of the common normal \vec{\mathbf{n}}_n and the main axis of the distal joint J_{n+1}. In the figure about common normals, these points are marked with red dots. If there is no unique common normal, so if the axes are parallel or collinear, the origin of the distal joint is chosen.
The last frame is always located in the center of the end-effector.
z-axis
The z-axis of coordinate frame K_n is always coincident with the main axis of joint J_{n+1}. This holds for the base frame as well as for all the intermediate frames. So the direction of the z-axes can easily be determined for the whole manipulator.
Corresponding to the notation, there is no joint and consequently no main joint axis at the end-effector. Thus the last frame K_k is usually kept parallel to the previous frame K_{k-1} if possible.
x-axis
The x-axis of frame K_n is defined as collinear to the common normal \vec{\mathbf{n}}_n of the link. This means that it lies on the same line, but could either have the same direction or the opposite. For simplicity, the direction of the x-axis should be kept as for the previous frame if possible.
As the base frame has no prior coordinate frame, there is no common normal \vec{\mathbf{n}}_0. Thus, if the world frame is not defined yet, the first x-axis is a free choice. Nevertheless it should be chosen well-considered as it is important for the determination of the Denavit-Hartenberg parameters.
For special case 2 of the common normal (intersection), the length of the common normal is 0, but its direction is distinct.
If the two joint axis are collinear (special case 3), the direction of the common normal is not distinct. As already mentioned before, in such a case, the direction of the x-axis should be kept preferably like before. This simplifies the determination of the Denavit-Hartenberg parameters.
Like mentioned before, the last coordinate frame K_k and so the x_k-axis is usually kept as before if possible.
Right-hand-rule.png
y-axis
As the x- and the z-axis are already defined, the y-axis can be determined using the right hand rule shown in the figure on the right.

For further illustration, watch the video in the next article about the Denavit-Hartenberg parameters. There the whole process including the assignment the coordinate frames and the determination of the parameters is explained very well.

Example: Assigning coordinate frames

The left side of the figure below shows a 5-link manipulator in its zero position and the corresponding main joint axes and common normals.

K_0
The base coordinate frame K_0 is already given located in the origin of joint J_1.
K_1
The red dot in J_1 indicates the intersection of the common normal with the main joint axis. So this is the origin of K_1. The z_1-axis is already determined by the main joint axis shown in blue. The x_1-axis is defined as collinear to the common normal. So it could either point into the figure or out of it. Like in other situations, the x-axis should always be kept as before if possible. So the x_1-axis is chosen pointing out of the figure parallel to x_0. Then following the right-hand-rule, the y_1-axis is directed downwards.
K_2
For J_2 and J_3 we have the parallel case of the common normal. So the location of the common normal is not unique. Thus the origin of coordinate frame K_2 is set to the origin of joint J_3, which is marked with a red dot. The z_2-axis is again already defined. But as the direction of the common normal is not unique, the x_2-axis is kept as before. So the y_2 stays as well as before.
K_3
The common normal of J_3 and J_4 is located in the intersection of the two joint axes (see red dot). So this is the origin of coordinate frame K_3. As you can see, the coordinate frames are not always located in the physical origins of the joints. The z_3-axis is defined by the main joint axis and the x_3-axis is defined as collinear to the common normal, so it can be kept as before. The y_2-axis is then pointing to the left.
K_4
The position of the common normal is actually not unique along the two axes (see example for the common normal). But in such cases the common normal is set such that it ends in the origin of the distal joint. So coordinate frame K_4 is located in the origin of J_5. The z-axis is already defined. As the common normal is now directed horizontally from the left to right, the x_4-axis has to be changed accordingly. It is to chosen to point to the right, although the opposite direction would also be correct. So the y_4-axis is directed into the figure.
K_5
The last coordinate frame is attached to the end-effector. As there are no restrictions to its orientation for this manipulator, the axes are kept parallel to the axes of K_4 to simplify the determination of the Denavit-Hartenberg parameters.
Dh-ex-frames.png