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 assigned to them first. There are several rules that have to be observed when assigning coordinate frames following the Denavit-Hartenberg convention.
Notation
The notation of the links and joints is shown in the figure below. A manipulator consists of links that are connected with joints. The links correspond to the rigid parts of an arm and the joints are the flexible connections between them. A joint is always assigned to the proximate link. So a link is connected with its joint to the end of link . The proximate link is then mounted to the end of via its joint . The first link is mounted on the base via joint . So the base is actually link but does not directly belong to the manipulator. The end of the last link corresponds to the end-effector.
The coordinate frames are always attached to the end of the links at the distal joints. The first coordinate frame, indexed , is the base or reference frame and attached to the base in joint . The next frame is at the end of link in joint followed by at the end of and so on. The coordinate frame of the last link is finally attached to the end of of the manipulator and so to the end-effector.
Main joint axes
Following the above 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. There is one rule, that is always valid. This is, that the -axis of a coordinate frame is always pointing in the direction of the main axis of the related joint. Like can be seen in the figure on the left, the main axis of a prismatic joint is the axis along which the displacement in positive direction is applied. For a revolute joint, the main axis is the rotation axis.