Abstract: For the under-constrainted system that multi-robots tow the payload by cables, which does not meet the condition of force closure and equations are generally not compatible, so the accurate workspace can't be found out. The static equilibrium equation of the system was established, and then the least squares principle combined with the Monte-Carlo method was put forward. When the movement is low speed or quasi static for the payload, the approximate static equilibrium workspace can be found out quickly and the expression of the cable tension is given by using the normal equations and QR decomposition respectively. When position of robots and position and pose of the payload happen that the wrench isn't failed, there is the unique least square solution.The effectiveness of the proposed method was validated by computer simulation, the method can be used to quickly find out static equilibrium workspace of the under-constrainted cable-driven system for different space configuration, which provides the foundation to the research of planning and control for the system.

Key words: under-constrainted, static equilibrium, workspace, simulation