Stable Kinematics for Multirobot Collaborative Transporting System With a Deformable Sheet

A deformable, flexible sheet that is held by multiple robots can be used for manipulating and transporting an object. For such multi-robot collaborative transporting system, forward kinematics provide object's potential positions where it might remain stationary on the sheet. Some of these forward kinematics solutions are however demonstrated unstable under object's position or robot formation perturbations. This paper presents stability criteria of the kinematics solutions of the object on the deformable sheet held by multi-robot system. We capture the sheet deformation and object position using the virtual variable cable model. A constrained quadratic problem is formulated to obtain the forward kinematics solution for a given multi-robot configuration. Linear dependence of active constraints and stability multipliers are used to assess the stability of the kinematics solutions. Two stability criteria are proposed and analyzed under object position or robot formation perturbations. These criteria are related to the properties and conditions of the stability multipliers. We present an efficient computational algorithm to determine stable kinematics, which are a small fraction of feasible solutions. Experimental results and case studies are presented to validate and demonstrate the effectiveness and efficiency of the analyses and algorithms.