Notation of links, joints and coordinate frames

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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.