In ARAS research groups, development of many robots such as hydraulic shoulder parallel robot, delta robot, wheeled mobile robot, cable driven redundant parallel manipulator, etc., has been carefully accomplished. Some of them like UAV and UGV are naturally underactuated, meaning that the number of actuators are less than the degrees of freedom. Moreover, cable driven robot in the suspended mode may be designed underactuated. Over recent decades, control of these systems is one of the attractive fields for researchers. Despite of several works in this field, there are some open problems that are the focus of our research. Underactuation leads to some constraints on configuration variables. In some cases velocity of robot is in a subset of overall space. These robots have velocity constraint and are called to have first order or velocity non-holonomic constraints. Examples of these robots are wheeled mobile robot and (nS)-2SP underactuated wrist robot. In other cases acceleration of robot is in a subset of space which leads to second order or acceleration nonholonomic. UAV and underactuated cable robot are examples of second order nonholonomic systems. There are some methods to control first order nonholonomic systems such as changing the equation of motion to chained form by which several robust and adaptive algorithm has been proposed for stabilization and trajectory tracking on them in literature. For controlling underactuated systems, Interconnection and Damping Assignment Passivity Based Control (IDA-PBC) is a common method. In this method a desired Hamiltonian structure is devoted to closed-loop. and then characterize all assignable energy functions compatible with this structure. This characterization is given in terms of the solution of a partial differential equations (PDEs) which are called matching condition. Therefore, one of the main objects of our team is to apply this method to two 3-DOF underactuated cable driven robots, solving corresponding PDEs and ensuring positive tension in cables.
