Adaptive feedback linearizing control of nonholonomic wheeled mobile robots in presence of parametric and nonparametric uncertainties
Introduction
The problem of motion control of Wheeled Mobile Robots (WMRs) is extensively studied in past few decades [1], [4], [5], [8], [10]. An important motion control problem is the trajectory tracking which is concerned with the design of a controller to force a WMR to track a geometric path with an associated timing law [7]. A variety of control algorithms for trajectory tracking problem are developed in the literature [14], [15], [17], [18], [20], [21]. Because of the challenging nonlinear model of WMRs, the feedback linearization technique is one of the successful design approaches to solve this problem. There are many works that propose tracking controllers based on feedback linearization for WMRs [2], [3], [6], [11], [14], [16], [27]. Campion et al. [27] investigated the controllability and feedback linearizability of the nonholonomic systems. Andrea-Novel et al. [13] applied the linearization technique to achieve tracking control of mobile robots. In [16], a tracking controller is proposed based on input–output feedback linearization for a nonsquare WMR system. Oriolo et al. [14] presented a design and experimental validation of dynamic feedback linearization to solve the trajectory tracking problem. However, most of them ignore the WMR dynamics in the design of controllers, which are inefficient for high speed massive WMRs. In addition, the works which propose the feedback linearization controllers for both kinematic and dynamic models of the WMRs mostly apply exact models and ignore their parametric uncertainties (for example, see [2]). This problem may cause not to achieve the exact cancellation of nonlinearities in the WMR model by input–output feedback linearization technique. Fortunately, adaptive control strategies present a reasonable solution to overcome parametric uncertainties. There are many key works to address the problem of tracking control of feedback linearizable systems [9]. However, the authors believe that the adaptive version of input–output feedback linearization control, as a powerful technique, has not been paid enough attention to solve the trajectory tracking problem of WMRs. Note that the main drawback of this control strategy is that the inversion of the estimate of decoupling matrix may not exist when the parameter estimates tend to zero. Therefore, this may lead to the divergence of tracking errors. This problem was solved by a technique, which restricts the parameters estimate to lie within some prior bounds [23], [26]. Another problem is that the integral-type of adaptive laws may lose their stability in presence of nonparametric uncertainties such as disturbances. Robust control strategies can modify the adaptive control law to overcome nonparametric uncertainties. This modification may be carried out in the design stage of the adaptation or control law.
The main contribution and novelty of the present work lies in designing an adaptive input–output feedback linearizing controller to solve the integrated kinematic and dynamic trajectory tracking problem of WMRs. The proposed controller is modified by: (1) a leakage modification on the parameter update rule to avoid parameters drift due to the nonparametric uncertainties, (2) a robust controller with an adaptive upper bounding function to compensate for the nonparametric uncertainties, which are motivated from the textbook of Lewis et al. [19] on the robot manipulators. Consequently, the formulation of the adaptive control law is also developed for the type (2, 0) WMR. Furthermore, in contrast to previous works, our proposed controller provides an actuator-level control signal from a practical viewpoint.
The rest of the paper is structured as follows. After a review of the kinematic and dynamic model of WMRs in Section 2, the tracking controller is proposed based on SPR-Lyapunov design approach in Section 3. The adaptive tracking controller is modified to be robust against nonparametric uncertainties in Section 4. Simulation results are presented for type (2, 0) WMR to illustrate the robustness and tracking performance of the proposed controllers in Section 5. Finally, conclusion and future works are presented in Section 6.
Section snippets
Kinematic and dynamic model of a nonholonomic WMR
In this section, we review a mathematical formulation of wheeled mobile robots with nonholonomic constraints which are moving on a planar surface. It is assumed that the configuration of the WMR is described by n generalized coordinates, q, subject to m constraints (m<n) as follows:where it includes k holonomic and m−k nonholonomic constraints, which all of them may be written in the form ofwhere is a full-rank matrix. Assume that S(q)=[
Controller design
A trajectory tracking control law can be designed based on adaptive feedback linearization technique for the WMR system as given in (15). The presented system in (15) might be summarized as the following affine MIMO nonlinear model:where x∈ℝn and f(x), q(x, θ), gi(x, θ) and ξ(x,t) are smooth vector fields on ℝn with g(0, θ)≠0. Remark 4 Based on the study of WMRs dynamics shown in (15), the following results might be summarized:
- 1.
The system is controllable and its
Modification for robustness
In practice, the WMR model is also subjected to nonparametric uncertainties which are described by Remark 3. Thus, we suppose that LξLfh(x)≠0 in (21). By following (26) through (31), one may rewrite (31) aswhere δ=LξLfh(x) denotes the input–output map of the uncertainty ξ(x,t) in the system (16). Then, by considering (32), (33), (34), (35), (36), (37), (38), (39), the entire system error equation might be re-written as
By differentiating (43) and substituting (49) in
Application to type (2, 0) WMR
The configuration of type (2, 0) WMR is shown in Fig. 2. The WMR has two conventional fixed wheels on a single common axle and a castor wheel to maintain the equilibrium of the robot. The centre of mass of the robot is located in PC=(xC,yC). The point P0=(xO,yO) is the origin of the local coordinate frame that is attached to the WMR body and is located at a distance d from PC. The point PL=(xL,yL) is a virtual reference point on x axis of the local frame at a distance L (look-ahead distance) of
Conclusion and future works
A trajectory tracking controller has been designed based on feedback linearization technique for nonholonomic wheeled mobile robots. The proposed controller can solve the integrated kinematic and dynamic tracking problem in presence of both parametric and nonparametric uncertainties. Following points were considered in the design process of the control law: (1) an adaptive law was designed based on SPR-Lyapunov approach to achieve robustness to parametric uncertainties. (2) It was shown that
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