A Geometrical Method for Calculating the Unreachable Workspace of the 3-DOF Gantry-Tau Parallel Manipulator

One of the main advantages of the Gantry-Tau machine is a large accessible workspace/footprint ratio compared to many other parallel machines. The Gantry-Tau improves this ratio by allowing a change of assembly mode without internal link collisions or collisions between the links and the moving TCP platform. This paper introduces the geometric approach for the unreachable workspace area for the Gantry-Tau. The unreachable area can occur in the middle of the workspace of reconfigurable PKMs with fixed length actuators. It is important to eliminate unreachable areas when designing the Gantry-Tau PKM because they appear in the middle of the workspace which is often the most useful part of the workspace. This approach is significantly faster than analytical workspace calculation methods based on the inverse kinematics. Because of the fast computational speed of the geometric approach, the method is ideal for inclusion in a design optimisation framework. The design presented in this paper achieves a workspace/footprint ratio of more than 2.7 with a zero unreachable workspace. Typical PKMs, for example the Delta robot, has a ratio less than one. In addition, a workspace optimisation method is presented where the parameters are the support frame lengths and the robot’s arm lengths.