Notation of links, joints and coordinate frames

← Back: Denavit-Hartenberg Convention

Overview: Denavit-Hartenberg Convention

Next: Types of joints →

The notation of the links and joints is shown in the figure below. A manipulator consists of $k$ links that are connected with $k$ 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 $L_n$ is connected with its joint $J_n$ to the end of link $L_{n-1}$. The proximate link $L_{n+1}$ is then mounted to the end of $L_n$ via its joint $J_{n+1}$. The first link $L_1$ is mounted on the base via joint $J_1$. So the base is actually link $L_0$ but does not directly belong to the manipulator. The end of the last link $L_k$ corresponds to the end-effector.

There are $k$ coordinate frames defined for the $k$ links of a manipulator plus one base or reference frame, respectively. 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 $K_k$ of the last link is finally attached to the end of of the manipulator and so to the end-effector.

Example: The notation of links, joints and coordinate frames of a manipulator The left side of the figure below shows a manipulator with $k=5$ links and joints. Further information about the joint types is provided in the next article. The joints are named $J_1$ to $J_5$ beginning with the first joint and the rigid parts attached to them are the links $L_1$ to $L_5$. At the end of the last link $L_5$ the end-effector is attached. Beginning with the base coordinate system $K_0$ (represented by the 3 axes $x_0$, $y_0$ and $z_0$) the coordinate frames $K_n$ are each attached to the joints $J_{n+1}$ and the last frame $K_5$ to the end-effector.