A single-loop spatial mechanism kinematically becomes an open robot, if we separate the grounded joint of the input link which may then be considered as the end effector of the robot. Any position of the end-effector within the workspace of such an open robot can be reached via a number of different configurations of the links. These configurations are called “branches” of the open robot for that particular position of the end effector.
If the open robot is now stretched to a limiting position by a force exerted on the end effector, all the possible branches of the mechanism approach each other. When they become coincident, they form the “limiting configuration”. Any two related branches are at opposite sides of the limiting configuration. From the relationship between the links in th elimiting configuration and in related branches, conditions for aviodance of branching of the original closed-loop mechanism can be obtained. This is necessary in order to assure that a set of consistent relative displacements are specified for the open robot to move displacements are specified for the open robot to move toward the desired end-effector position without jumping from one branch to another. As for the closed-loop mechanism, open robot branching aviodance ensures that a desired sequence of positions of a particular floating link in the loop will be generated without changing the branch of the link configuration.
In this paper, the above approach is applied to RSSR, RRSC, RRSRR, RRRRRRR and RPCRRR spatial closed-loop motion-generator mechanisms and the corresponding conditions for aviodance of branching in the synthesis of the mechanisms are derived.