Applications of the Block Backstepping Algorithm on Underactuated Mechanical Systems with Higher Degrees of Freedom: Some Case Studies

Abstract

Applications of the proposed control algorithm on 2-DOF underactuated mechanical systems (UMS) have already been discussed in the previous chapter. Needless to say that during real-life applications, most of the practical UMSs that comes in the scenario has more degrees of freedom. Therefore, only dealing with the 2-DOF systems would not give readers full working knowledge. Keeping in view the immense importance of higher order UMSs, authors dedicate this chapter to describe the applications of the proposed control algorithm on the same in a systematic manner. Following the similar presentation approach of Chap. 4, at the onset, a very simple flat 3-DOF system model is considered to demonstrate the controller design procedure for higher order systems. Since flat UMS has most simple dynamic characteristics than that of the other UMSs, it is always easy to deal with the control problems for such type of systems. Application of the proposed control algorithm on a vertical-takeoff-landing air craft (VTOL), which is also a type of flat UMS, is described in the first section. Thereafter, Sect. 5.2 describes application of control law on underactuated surface vessel (USV). Being a member of nonholonomic systems, it fails to satisfy the Brocket’s condition of feedback linearization. Therefore, USV requires nonsmooth or time varying control input for its stabilization. Needless to say that designing a control law for USV is more difficult than that of other holonomic UMS (e.g., VTOL). Nonetheless, without considering robotic applications, discussions on UMSs would not be able to take its complete shape. Therefore, at the end, Sect. 5.3 demonstrates application of the control law on the robotic manipulator. Like USV, 3-DOF manipulator also belongs to the class of nonholonomic systems; however, unlike USV it possesses interacting control inputs. Hence, it is easy to understand that control law design for the same is more difficult than that of USV. Like the previous chapter, here also the reader will observe that no such significant modification is required to recast the control law for individual systems. Proposed control law is generalized enough that it can address the control problems of most of the higher order UMSs. All being well after going through this chapter, the readers will find themselves ready to design control law for any practical underactuated systems.