Kinematic analysis of the 4-3-1 and 3-2-1 wire-driven parallel crane

The kinematics of wire-driven parallel robot (WDPR) is one of the most challenging problem of modern kinematics. We investigate here the kinematics of two robots with 3 and 4 wires, in a crane configuration, having specific location of the anchor point on the platform. The 4-3-1 robot has 4 wires, three of them being connected at the same anchor point on the platform. We show that this robot has up to 4 solutions to the forward kinematics. The 3-2-1 robot has 3 wires, two of them being connected at the same point on the platform: we show that the forward kinematics solutions of this robot, that involves its static equilibrium, may be obtained by solving a polynomial of degree 64, leading to a maximum of 256 solutions to the FK. In both cases the proposed maximum number of solutions is much better than the one already known for robots of general structure having 3 or 4 wires. Numerical experiments let us conjecture that the 64th order polynomial will always factor out and that all roots will be obtained from a polynomial of degree 32, leading to at most 128 solutions and we have obtained examples with up to 10 solutions with positive tensions in the wires. We justify this study by experimentally showing that a 4-3-1 robot moving in its workspace may become a 3-2-1 robot as one wire become slack, numerical simulations showing that this will happen in more than 50% of the poses in the workspace (but the 4-3-1 has obviously a larger workspace than the 3-2-1, hence its interest).