Robust Modeling and Control Issues of Parallel Manipulators with Actuation Redundancy

Parallel manipulators, often called parallel kinematics machines (PKM), are controlled non-linear dynamical systems. From a mechanical point of view PKM are (holonomically) constrained mechanical systems characterized by a power transmission between input and output. A so-called end-effector (EE), representing the mechanical output, is connected to a fixed platform by several (often identical) serial linkages, and the constraints reflect the existence of closed loops formed by these chains. Each chain is equipped with one or more actuators, representing the mechanical inputs. The modeling, identification, and control of PKM have advanced in the last decades culminating in successful industrial implementations. Yet the acceptance of PKM is far beyond that of the well-established serial manipulators. This is mainly due to the limited workspace, drastically varying static and dynamic properties, the abundance of singularities within the workspace, and the seemingly complex control. Traditionally the number of mechanical inputs of a PKM equals its mechanical degree-of-freedom(DOF) so that the PKM is non-redundantly actuated. Ameans to overcome the aforementioned mechanical limitations is the inclusion of additional actuators, commonly by adding further limbs to the moving platform without increasing the DOF of the PKM. As a simple example consider the PKM in figure 1. The EE can be positioned in the plane thus possesses 2 degrees of freedom. Also the PKM as a whole has the DOF 2 so that two actuators would be sufficient for controlling this PKM. Yet the PKM is actuated by 3 actuators, which gives rise to actuation redundancy in the sense that the actuator forces are not independent. Such actuation redundancy has the potential to increase the EE-acceleration, to homogenize stiffness and manipulability, and to eliminate input singularities (where the motion of the moving platform is not controllable by the actuators), and thus to increase the usable workspace. The design of RA-PKM, and the possible dexterity improvement were addressed in several publications as for instance Garg et al. (2009); Gogu (2007); Krut et al. (2004); Kurtz & Hayward (1992); Lee et al. (1998); Nahon & Angeles (1989); O’Brien & Wen (1999); Shin et al. (2011); Wu et al. (2009). The existence of redundant actuators allows for control forces that have no effect on the PKM motion but rather lead to mechanical prestress within the PKM. This effect can be exploited for different second-level control tasks such as backlash avoidance and stiffness control. In