Block Backstepping Control of the Underactuated Mechanical Systems

Abstract

Underactuated mechanical system [UMS] is a particular class of a multi output mechanical system that has more degrees of freedom than that of control inputs. According to Spong “Underactuated mechanical systems have fewer control inputs than degrees of freedom and arise in applications, such as space and undersea robots, mobile robots, flexible robots, walking, brachiating, and gymnastic robots” (Spong in Control problems in robotics and automation. Springer, Berlin, pp 135–150, 1998, [5]). Nonetheless, the complicated state model of this class of systems often makes the controller design tasks more difficult than that of the ordinary MIMO systems (Yu and Liu in IET Control Theory Appl 7:921–935, 2013, [6]). Comprehensive research on existing literature reveals that all the proposed approaches are either too complicated for practical implementation and only capable of ensuring guaranteed performance during theoretical analysis, or they are just apt for a particular application that reduces their scope of applicability on other control problems of similar nature (Rudra et al.). All these shortcomings of the previously proposed approaches had strongly motivated the authors to conceive an alternate control approach for achieving a tradeoff between theory and practice. However, while the authors were looking for a suitable control law that could be utilized to serve the said purpose, they observed different features of backstepping could help them to reach their objective. After that they toiled for a period of eighteen months, around the clock, and eventually they have devised a block backstepping control law, which is generalized enough to address the control problem of most of the UMS. Indeed, the proposed control law has initially been formulated to address only the stabilization problem of the UMS; even so, during further research it has been revealed that it can also be used to solve the tracking control problem for the same. This chapter presents a comprehensive description of the control law together with some illustrations so as to make this book an easy-reading one. For the ease of understanding, at first, the proposed algorithm is described for 2-DOF UMS, and thereafter generalized version of the proposed control algorithm is presented to address the control problem of n-DOF UMSs.