Abstract
In this paper, a generalized nontriangular normal form is presented to facilitate designing a recursive integral backstepping control for the class of underactuated nonholonomic systems, i.e., wheeled mobile robots (WMRs) that perform posture stabilization and trajectory tracking in environments without obstacles. Based on the differential geometry theory, we develop a multiple input multiple output (MINO) generalization of normal form using the input-output feedback linearization technique. Then, the change of variables (diffeomorphism) transform the state-space model of WMR, incorporating both kinematic and dynamic models into nontriangular normal form. As a result, the system dynamics can be represented as internal and external dynamics. The nonlinear internal dynamics of WMR pose serious challenges to design a suitable controller due to its internal dynamics being not minimum phase and non-strict feedback form structure. The proposed backstepping controller is designed in two steps. First, a standard integral backstepping controller is designed to stabilize the robot’s orientation angle. Then, a recursive integral backstepping control technique is applied to achieve asymptotic convergence of position error to zero. Hence, both asymptotic posture stabilization and trajectory tracking are achieved in semi-global regions, except the nonzero initial condition of the orientation angle. The asymptotic stability of the entire closed-loop system is shown using the Lyapunov criteria.
Highlights
Over the last decade, feedback control for the class of mechanical systems that possess both underactuated and nonholonomic behavior has gained remarkable interest among control researchers
It should be pointed out that three main reasons can be summarized that hinder the achievement of posture stabilization using the input-output feedback linearization approach: (1) The wheeled mobile robots (WMRs) can not be input-state linearizable by a smooth feedback control due to nonholonomic constraint [1,15], (2) internal dynamics of WMR are not minimum phase [1,16], and (3) the underactuated nonholonomic system (WMR) provides nontriangular normal form the structure [29]
The results show the posture stabilization of WMR to origin from the initial posture
Summary
Feedback control for the class of mechanical systems that possess both underactuated and nonholonomic behavior has gained remarkable interest among control researchers. It should be pointed out that three main reasons can be summarized that hinder the achievement of posture stabilization using the input-output feedback linearization approach: (1) The WMR can not be input-state linearizable by a smooth feedback control due to nonholonomic constraint [1,15], (2) internal dynamics of WMR are not minimum phase [1,16], and (3) the underactuated nonholonomic system (WMR) provides nontriangular normal form the structure [29]. This paper proposes a generalized nontriangular normal form, as a special class of the Byrnes–Isidori normal form introduced in [40], to facilitate the recursive integral backstepping control for the class of MIMO underactuated nonholonomic system, i.e., WMR to solve trajectory tracking and posture stabilization. This paper proposes a novel recursive integral backstepping control based on generalized nontriangular normal form structure for differential drive WMR.
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