Assigning coordinate frames

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Overview: Denavit-Hartenberg Convention

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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 $n$ links that are connected with $n$ 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 $L_i$ is connected with its joint $J_i$ to the end of link $L_{i-1}$. The proximate link $L_{i+1}$ is then mounted to the end of $L_i$ via its joint $J_{i+1}$. The first link $L_1$ is mounted on the base via joint $J_1$. So the base is actually link $L_0$ that does not directly belong to the manipulator. The end of the last link $L_n$ 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 $K_0$, is the base or reference frame and attached to the base in joint $J_1$. The next frame is $K_1$ at the end of link $L_1$ in joint $J_2$ followed by $K_2$ at the end of $L_2$ and so on. The coordinate frame $K_n$ of the last link is finally attached to the end of of the manipulator and so to the end-effector.