On the Force Control of Kinematically Defective Manipulators Interacting With an Unknown Environment

In this paper, the problem of force feedback control of kinematically defective manipulators (KDMs) is considered. KDMs are robot manipulators that have fewer joints than the dimension of the space in which their end-effector moves. It is well known that controlling the end-effector velocity of an $n$-joint KDM can be easily solved by appropriately selecting $n$ components of the output twist, thus squaring the control problem. On the contrary, we show that such a component selection approach is not appropriate in general to solve the force control problem for KDMs. In particular, for advanced force control applications, such as comanipulation, where the contact geometry is not known in advance, the selection of the wrench components leads to a lack of passivity, which in turn may induce instability. This instability does not arise from the system dynamics. Rather, it can be viewed as a new form of kinematic instability. Moreover, by formulating the problem in the joint space, we show how to properly design a stable force controller for KDMs subject to arbitrary external forces applied to their end-effector. Furthermore, we propose several implementations for pure force control and damping control. Experimental results with a kinematically defective laparoscopic comanipulator illustrate these propositions.