Finite-Time Stabilization of Dynamic Nonholonomic Wheeled Mobile Robots with Parameter Uncertainties

1 Mathematics and Physics Department, Hohai University, Changzhou Campus, Changzhou 213022, China 2 College of Mechanics and Materials, Hohai University, Nanjing 210098, China 3 State Key Laboratory of Hydrology-Water Resources and Hydraulic Engineering, Hohai University, Nanjing 210098, China 4Department of Control Science and Engineering, University of Shanghai for Science and Technology, Shanghai 200093, China 5 Department of Mathematics, Shijiazhuang University, Shijiazhuang 050035, China


Introduction
Stabilization problem of nonholonomic systems is theoretically challenging and practically interesting.As pointed out in [1], although every nonholonomic system is controllable, it cannot be stabilized to a point with pure smooth (or even continuous) state feedback law.In order to overcome the difficulty of Brocket's condition [1], a variety of sophisticated feedback stabilization methods have been proposed which mainly include continuous time-varying feedback control laws [2][3][4], discontinuous feedback control laws [5][6][7][8], and hybrid feedback control laws [9].
A common characteristic of these designs of controllers above is based on kinematic model, where only a kinematic model is considered and the velocities are taken as control inputs.But in fact, for some mechanical systems with nonholonomic constraints, it is more realistic to formulate the control problems at dynamic levels, where the torque and force are chosen as new inputs.Some results can be found in recent papers, for example, the dynamic tracking control of wheeled mobile robots in the presence of both actuator saturations and external disturbances is considered in [10], where a computationally tractable moving horizon H ∞ tracking scheme is presented.In [11,12], the saturated stabilization and tracking control are discussed for simple dynamic nonholonomic mobile robot.For uncertain dynamic nonholonomic systems, Ma and Tso [13] have given a robust control law for the exponential regulation of an uncertain dynamic nonholonomic wheeled mobile robot, in which the authors improved the convergence speed of regulating the state to a desired set point for the first time.
In order to drive a system to the equilibrium point with a fast convergence rate, finite-time stability theory has become a studying focus recently, for example, finitetime stabilization problems have been studied mostly in the contexts of optimality, controllability, and deadbeat control for several decades [14][15][16].Compared to the asymptotic stabilization, the finite-time stabilization, which renders the trajectories of the closed-loop systems convergent to the origin in a finite time, has many advantages such as fast response, high tracking precision, and disturbance rejection properties.
For the nonholonomic systems, a few researchers have got some excellent results in finite-time control field.In [17], the relay switching technique and the terminal sliding mode control scheme with finite-time convergence are used for the design of the controller to address the tracking control of the nonholonomic systems with extended chained form.For a class of uncertain nonholonomic chained form systems, Hong et al. [18] have designed a nonsmooth state feedback law such that the controlled chained form system is both Lyapunov stable and finite-time convergent within any given settling time.And the finite-time tracking control for single mobile robots or multiple nonholonomic mobile robots is considered in [19][20][21].The previous developed controller for nonholonomic systems can be divided into two categories: one is for the finite-time stabilization problem of chained form systems and the other is for the tracking control problem of mobile robots.However, to the best of our knowledge, there exist no results to deal with the robust finite-time stabilization of uncertain dynamic nonholonomic mobile robots.
This paper considers the stabilization problem of dynamic nonholonomic mobile robots with uncertain parameters in a finite time.The main results and contributions can be summarized as the following two respects.
(a) An uncertain 5-order chained form system can be obtained under the equivalent coordinate transformation of states, which means the finite-time stabilization of the chained form system is equivalent to the finite-time stabilization of the original dynamic robot system.
(b) Applying the theory of finite-time stability and the switching control method, we design a discontinuous robust controller to make the states of the chained form system converge to the equilibrium point in a finite time.
The structure of the paper is as follows: Section 2 gives a formalization of the problem considered in the paper.Section 3 states our main results.Section 4 provides an illustrative numerical example and the corresponding simulation results of the proposed methodology.Finally, a conclusion is shown in Section 5.

Problem Statement
A class of nonholonomic wheeled mobile robots are shown in Figure 1, the two fixed rear wheels of the robot are controlled independently by motors, and a front castor wheel prevents the robot from tipping over as it moves on a plane.Assuming that the geometric center point and the mass center point of the robot are the same and that the radiuses  are identical for all the rear wheels, 2 is the length of the fixed two rear wheels, where  and  are known positive constants.Its kinematic and dynamics model can be described by the following differential equations [13]: where ,  are the position of the mass center  of the robot moving in the plane and ,  are the mass and inertia of mobile robots; respectively, V is the forward velocity,  is the steering velocity and  denotes its heading angle from the horizontal axis, and  1 ,  2 are driving torques on the right and left rear wheels.
As pointed out in [11], take an orthogonal coordinate transformation Then system (1) can be converted to the following equation: where  1 =  1 −  2 ,  2 =  1 +  2 are new control inputs and  1 = /,  2 = 1/ are new unknown parameters with their bounds derived from (2) as follows: Because the coordinate transformation ( 3) is globally invertible and does not change the origin, it is obvious that the equilibrium point ( 1 ,  2 ,  3 ,  4 ,  5 ) = (0, 0, 0, 0, 0) is finitetime stable for its closed-loop system of (4) it is means that (, , , V, ) = (0, 0, 0, 0, 0) is also a finite time stable equilibrium point for the corresponding closed-loop system of (1).Hence, the control task is to design a discontinuous finitetime stabilizing controller for system (4) with the unknown parameters (5).Here, it should be noted that Hong et al. have designed a switching control strategy to discuss the finitetime stabilization of uncertain chained form systems in [18], however, it is invalid to control the dynamic chained system (4); thus, a new improved discontinuous design method is required.

Main Results
In this section, the main results will be presented.Firstly, we will state the basic idea to design a finite-time switching controller for system (4).Note that system (4) can be decoupled into two subsystems, one of which is { 1 ,  4 }-subsystem and the other describes the rest of (4), that is, { 3 ,  2 ,  5 }subsystem By designing  1 , the state  4 of ( 13) can be driven to any predetermined point in a finite time, based on which ( 3 ,  2 ,  5 ) of ( 14) can be stabilized to (0, 0, 0) by designing the finite-time controller  2 , and the last step is to redesign  1 such that ( 1 ,  4 ) can be driven to (0, 0) in a finite time.
The following are the switching controller design and the corresponding proofs of Theorems 4 and 6.
Then system (4) can be stabilized to the origin equilibrium point in a finite time by the switching controller Step 1-Step 3.
This completes the proof of the theorem.
Remark 5. Then control objective can be completed in each step within a finite time, and thus system (4) can be stabilized to zero in a finite time.
On the other hand, from (3), we have the following: Therefore, the finite-time switching controller for the original robot system (1) can be stated as follows.
Then system (1) can be stabilized to the origin equilibrium point in a finite time by the switching controller Step 1  -Step 3  .
On the other hand, the switching controller ( 1 ,  2 ) in Theorem 4 can be used to stabilize the states ( 1 ,  2 ,  3 ,  4 ,  5 ) of system (4) in a finite time; hence, the control task is changed to find the relation between the original controller ( 1 ,  2 ) of system (1) and the controller ( 1 ,  2 ) of system (4).
Comparing system (4) with system (1), we have the following: from which, and by using the switching controller of Step 1- Step 3 in Theorem 4, we can solve the corresponding ( 1 ,  2 ) and thus the conclusion can be obtained in Theorem 6.This completes the proof of the theorem.

Simulations
In this section, the discontinuous switching controller proposed in theorems above is used to show how to stabilize the state ( 1 ,  2 ,  3 ,  4 ,  5 ) of ( 4) and the state (, , , V, ) of (1) to the zero equilibrium point in a finite time.We will demonstrate the effectiveness of our methods by a numerical example.
Figures 2-5 show some simulation results with Matlab.From Figures 2 and 3, it can be seen that all the state variables of the closed system (4) are driven to the origin equilibrium point in a given settling time  3 = 30 s. Observing Figure 2, in time interval 0∼5 s, the first step control task is completed, that is,  4 =  0 = 1 as  ≥ 5 s.Next, from Figure 3, it is clear that ( 3 ,  2 ,  5 ) can be stabilized to zero by the controller  2 in Step 2 as  ≥ 20 s and remain unchanged.Finally, the controller  1 in Step 3 drives ( 1 ,  4 ) to zero in the settling time  ≤  3 .Figures 4 and 5 show the finite-time convergence of the state variables (, , , V, ) of the original robot system (1) exactly as the state of the transformed system.

Conclusion
The robust finite-time stabilization problem is discussed in this paper for a class of uncertain dynamic nonholonomic wheeled mobile robot.The contributions of this paper include having applied finite-time control technique and a new switching design method such that all the states can be stabilized to the zero point by the proposed discontinuous  controller.And we will work on extending the results to consider the corresponding trajectory tracking control problem in the coming time.