Stochastic Stabilization of Nonholonomic Mobile Robot with Heading-Angle-Dependent Disturbance

The problem of exponential stabilization for nonholonomic mobile robot with dependent stochastic disturbance of heading angle is considered in this paper. An integrator backstepping controller based on state-scaling method is designed such that the state of the closed-loop system, starting from a nonzero initial heading angle, is regulated to the origin with exponential rate in almost surely sense. For zero initial heading angle, a controller is designed such that the heading angle is driven away from zero while the position variables are bounded in a neighborhood of the origin. Combing the above two cases results in a switching controller such that for any initial condition the configuration of the robot can be regulated to the origin with exponential rate. The efficiency of the proposed method is demonstrated by a detailed simulation.


Introduction
In the past decades, there has been increasing attention devoted to the control of nonholonomic systems such as knife edge, rolling disk, tricycle-type robot, and car-like robot with trailers see, 1, 2 and the references therein .From Brockett's necessary condition 3 , it is well known that the nonholonomic systems cannot be stabilized to the origin by any static continuous state feedback, so the classical smooth control theory cannot be applied directly.This motivates researchers to seek for novel approaches such as discontinuous feedback and time-varying feedback.The discontinuous feedback uses the state-scaling technique and switching control strategy 4, 5 , which usually results in an exponential convergence.The time-varying feedback provides smooth controllers, but its convergence rate usually is slow 6, 7 .All the above references considered the nonholonomic systems in the deterministic case, while the nonholonomic systems with stochastic disturbance have rarely been researched up to now.

Mathematical Preliminaries
Consider the nonlinear stochastic system dx f x, t dt g x, t dW, x t 0 x 0 ∈ R n , 2.1 where x ∈ R n is the state, f 0, t 0, g 0, t 0, and W is an r-dimensional independent standard Wiener process.
The following notion of boundedness on an interval in probability can be seen as a slight extension from that used in 9 .
Definition 2.1.A stochastic process x t is said to be bounded on t ∈ t 0 , T , where T ≤ ∞, in probability if the random variable |x t | satisfies lim R → ∞ sup t∈ t 0 ,T P {|x t | > R} 0.

2.2
For this notion, a corresponding criterion can be easily obtained following the line of 10 .
Lemma 2.2.Consider system 2.1 defined in t 0 , T , where T ≤ ∞.Assume that there exist a function V ∈ C 2 , class K ∞ functions α |x| and α |x| , a positive constant c, and a nonnegative constant d such that for all d, for all t ∈ t 0 , T , then system 2.1 has a unique solution on t 0 , T , which is bounded on t ∈ t 0 , T in probability.
To find condition to let state scaling make sense, the following lemma proved by Mao in 11, pages 51, 120 is recited as follows.
Lemma 2.3.For system 2.1 defined on t ∈ t 0 , T , where T ≤ ∞, assume that there exist two constants K 1 and K 2 such that i (lipschitz condition) for all x, y ∈ R n and t ∈ t 0 , T then there exists a unique solution x t : x t 0 , x 0 , t to system 2.1 and for all x 0 / 0 in R n , P x t, t 0 , x 0 / 0 1, ∀T ≥ t ≥ t 0 2.5 (i.e., almost all the sample path of any solution starting from a nonzero state will never reach the origin).
The concepts of moment exponential stability and almost surely exponential stability together with their criteria can be found in 12, page 166 , which are presented here for selfsufficiency.
Definition 2.4.For p > 0, system 2.1 is said to be pth moment exponential stable if lim sup

Problem Formulation
A nonholonomic mobile robot of tricycle type in the presence of stochastic disturbance can be described by where u is the forward velocity, v is the steering velocity, x c , y c is the position of the mass center of the robot moving in the plane, θ is the heading angle from the horizontal axis, W 1 and W 2 are two independent standard Wiener processes, and ϕ 1 and ϕ 2 are two unknown scaler-valued smooth functions.A tricycle-type robot is described by Figure 1.
Performing the change of coordinate x 1 θ, x 2 x c sin θ − y c cos θ, x 3 x c cos θ y c sin θ, 3.2 system 3.1 can be transformed into system 1.1 .The control objective is to design a state-feedback controller such that all the signals in the closed-loop system are globally exponentially regulated to the origin in probability.For this end, the following assumptions are imposed throughout this paper.A1 There exists a positive constant k such that A2 There exist a positive constant l and a smooth nonnegative function φ such that

3.4
Remark 3.1.System 1.1 is similar to the class of systems in strict-feedback form driven by Wiener processes, which motivates us to investigate the backstepping controller design that had been extensively researched by 8, 13 .Assumptions A1 and A2 are given to diffusion terms as same as those imposed to drift terms in 5 in the deterministic case.For nonzero initial value of x 1 , by imposing A1 on function ϕ 1 , the state x 1 can be regulated to zero with exponential rate but never reach zero see the subsequent subsection , which is the key to introduce a state-scaling transformation to deal with other troubles see Section 1 .

Design of Controller v
It can be seen that the state x 1 of system 1.1 can be globally exponentially regulated to zero via a static feedback control law.In fact, we can introduce a Lyapunov function By choosing the control law v as where λ ≥ 2k 2 is a positive parameter further requirements for λ will be given later , 4.2 becomes By substituting 4.3 into the first equation of 1.1 , one has which, together with assumption A1 and Lemma 2.3, means that there exists a unique solution to 4.5 and that any solution starting from a nonzero state will never reach the origin in almost surely sense.From assumption A1 , 4.1 and 4.4 , according to Lemma 2.3, the solution exponentially converges to zero, that is, lim sup t → ∞ 1/t log |x 1 t | < −λ, a.s., which means that

State-Scaling Transformation
We have designed controller v such that state x 1 t can be globally exponentially regulated to zero.Consequently, v will converge to zero as t goes to ∞.This causes trouble in the control of x 2 -subsystem and x 3 -subsystem.To overcome this difficulty, we introduce a state-scaling transformation defined by According to the comment in the end of Section 1, the transformation 4.7 makes sense in almost surely sense, for the initial value x 1 t 0 / 0. From 1.1 , 4.3 , and 4.7 , we have 4.8

Backstepping Controller Design of u
In this part, controller u will be constructed, based on backstepping techniques, under the assumption x 1 t 0 / 0.

Mathematical Problems in Engineering 7
Step 1. Begin with z 1 -subsystem of 4.8 , where x 3 is regarded as a virtual control.Introducing the transformation and choosing Lyapunov function it comes from 4.8 -4.10 that Here, the terms α and α 2 appear in the same time, which is different from the traditional backstepping procedure.Considering assumption A1 and the characters of terms of 4.11 , the virtual control is chosen as where c 1 > 0 is a design parameter.By the aid of 4.4 , 4.12 , and ϕ 1 /x 1 2 ≤ k 2 that comes from assumption A1 , 4.11 can be rewritten as

4.13
Submitting the inequalities

4.14
where d > 0 is a design parameter, into 4.13 gives

4.16
Step 2. In view of 1.1 and 4.9 , we have

4.17
Consider the candidate Lyapunov function whose infinite generator along 4.17 satisfies

4.19
By using Young's equality and A1 , A2 , 4.7 , and 4.9 , it is easy to obtain that

4.20
Mathematical Problems in Engineering 9 which are submitted into 4.19 to give

4.24
Summing up all the requirements to λ leads to

4.25
Remark 4.1.It is noteworthy that the terms in control u is separated into two groups.The terms caused by the state scaling are put in u 2 , in other words, if the transformation z 1 x 2 /x 1 is replaced with nonscaling one z 1 x 2 , the terms in u 1 will still remain in u.This will be used in the subsequent section.

Stability Analysis
It is position to give stability conclusion for the case of x 1 t 0 / 0.

Controller Design to Drive x 1 Away from Zero
Considering the transformation z 1 x 2 /x 1 , the control u given by 4.22 will escape to infinite for an initial state with element x 1 t 0 0. The first thing before the controller 4.22 does work is to drive the state x 1 t away from zero in a small distance denoted by r.For a given r > 0, define a stopping time τ r inf{t : t ≥ t 0 , |x 1 t | ≥ r}.To let the state x 1 leave zero, the control v can be chosen as v −λ 5.1 during t 0 , τ r , where λ is the same design parameter used in 4.25 some explanation will be given latter .In this case, the x 1 -subsystem becomes

5.3
The existence and uniqueness of solution of x 1 -subsystem in t 0 , τ r comes from assumption A1 and Lemma 2.3.Since during the interval t 0 , τ r , the controller 4.22 cannot be used.A new scheme for u is expected to bound the states x 2 and x 3 in a neighborhood of the origin in this interval when x 1 is being driven away from the origin.Substituting v −λ into the last two equations of 1.1 gives Since 5.4 is a standard strict-feedback form, viewing x 1 as an external bounded input, backstepping controller can be designed to make the states x 2 and x 3 to be bounded in probability in t 0 , τ r .Introduce the transformation which implies that where α c 1 z 1 is used as in 4.12 with a design parameter c 1 > 0. A careful observation indicates that all the terms in 5.6 have the corresponding terms in 4.8 and 4.17 , that is, if ϕ 1 /x 1 in the latter is replaced with ϕ 1 , then we can obtain the terms in the former.
In t 0 , τ r , we have |x 1 | ≤ r.To design controller u to guarantee the boundedness of x 2 and x 3 , a candidate Lyapunov function is given as follows:

5.7
Just for simplicity, we will design the controller as consistent as possible with statescaling case in the proceeding section.By selecting r ≤ 1, according to assumption A1 , we can see that in the nonscaling case, we have ϕ 2 1 ≤ k 2 r 2 ≤ k 2 , which is corresponding to ϕ 1 /x 1 2 ≤ k 2 used in state-scaling case 4.13 .Comparing 5.5 -5.7 with the corresponding equalities in the proceeding section, it can be found that, in the nonscaling case, some terms used in 4.22 that are included in u 2 disappear and the others that are contained in u 1 have the same forms with the same or milder requirements to the parameters c 1 and λ.Therefore, by choosing u u 1 , 5.8 where d c 3/4 l 2 r 4 and c min{c 1 λ 1 − e , 4c 2 }, which implies that

5.10
The stability analysis before τ r can be included in the following result.
Theorem 5.1.Under assumptions (A1) and (A2), for every x 1 t 0 0 and any x 2 t 0 , x 3 t 0 , for any 0 < r ≤ 1, with an appropriate choice of the design parameters λ and c 1 , the closed-loop system consists of 3.1 , 3.2 , 5.1 , and 5.8 has a unique solution, and all the signals are bounded in probability in the interval t 0 , τ r .
Proof.According to Lemma 2.2, the existence and uniqueness of z 1 and z 2 in t 0 , τ r come from 5.7 and 5.9 .Noting the existence and uniqueness of x 1 , 4.7 , 4.9 , and 4.12 , the existence and uniqueness of x 2 and x 3 can be concluded on t 0 , τ r .Following the same line, the boundedness of x i i 1, 2, 3 on t ∈ t 0 , τ r can be obtained, which complete the proof.

Design of Switching Controller
Since |x 1 τ r | r > 0, at the stochastic moment t τ r , we switch the control laws v and u from 5.1 and 5.8 to 4.3 and 4.22 , respectively.According to Theorem 4.2, the solution of the closed-loop system converges to the origin with exponential rate on τ r , ∞ for any r > 0. A switching control scheme on t 0 , ∞ can be given as where the switching signal is defined by 1, t ∈ τ r , ∞ .

6.2
By summing up the above arguments, the main result in this paper can be presented now.
Theorem 6.1.Under assumptions (A1) and (A2), for x 1 t 0 0 and any x 2 t 0 , x 3 t 0 , with an appropriate choice of the design parameters λ, c 1 and r, the closed-loop system consists of 3.1 , 3.2 , and 6.1 has a unique solution on t 0 , ∞ , which is 4th moment exponential stable.
Proof.For any 0 < r ≤ 1, the existence of a.s.finite stopping time τ r can be concluded from 5.3 .The existence and uniqueness of solution of the closed-loop system can be proved by Theorem 5.1 on t 0 , τ r and by Theorem 4.2 on τ r , ∞ , respectively.In the interval t 0 , τ r , there holds |x 1 | ≤ r, and from 5.10 , we have EV z 4 1 t z  A new question about the performing of switching signal comes forth.One scheme is presented as follows in a discrete-time form.Suppose that the running time interval is 0, T and every step equals to Δ T .Initial step: begin with t 0, τ 1 T , and τ 2 0. Recursive steps: perform the following procedures in turn unless otherwise stated.a Write down the value of x x t and let t t Δ. b If t > T, then turn to g , otherwise, perform the following calculation.c If t ≤ τ 1 , then we have t m t, otherwise, we have t m τ 1 ; we have s 0, τ 1 τ 1 and restore τ 2 t, otherwise, we have s 1, τ 2 τ 2 and τ 1 t m .e Submitting s into control 6.1 and resolve the response x t of closed-loop system.f Turn to a .g Output the observed value τ r τ 2 .h End the procedure.The procedure is described in Figure 2.
It should be pointed out that the switching strategy will lead to trembling phenomenon.In practice, to eliminate the trembling, the switching signal given by 6.2 can be replaced by a continuous one which depends on the measurement of τ r .The above logic method will be used in the forthcoming simulation.

Simulation
Consider system 3.1 with ϕ 1 kθ and ϕ 2 lθ 2 .By letting φ θ θ, the assumptions A1 and A2 can be easily verified.As pointed out by 14, page 63 , system 3.1 is an idealization of the following system: with white noises N 1 and N 2 , which is formally obtained by replacing "dW i t /dt" by N i t .
To give approximate simulation using ordinary differential equation algorithm, system 3.1 is replaced by 7.1 , where the power of each N i equals to 1.
The following two cases are to be analyzed: 1 θ 0 −1.5, x c 0 0.8, y c 0 −1, k 0.1 and l 1; 2 θ 0 0, x c 0 0.8, y c 0 −1, k 0.1 and l 1.For the first case, the state-feedback control law is given by 6.1 not 4.3 and 4.22 with the design parameters d 0.8, e 0.7, c 1 1/e 1 3/4 d , c 2 3, r 1, and λ satisfying the equality of 4.25 .Figure 3 demonstrates that the state of the closed-loop system can be regulated to the origin with exponential rate in almost surely sense without switching.For the second case, the same control 6.1 with the same design parameters as in the first case is given.From Figure 4, we can see that switching happens at the moment τ r ≈ 0.1258.The state of the closed-loop system can be driven to the origin with exponential rate after moment τ r in almost surely sense .To eliminate the trembling phenomenon, the switching signal s given by 6.2 can be replaced by a continuous one.Figure 5 describes the responses of the closed-loop system of Case 2 with the following s t : t ∈ 0, τ r , 2 π arctan 600 t − τ r , t ∈ τ r , ∞ .

7.2
Comparison of Figure 4 with Figure 5 indicates that control magnitude in the latter is milder than that in the former.

Conclusions
A global exponential stabilization controller has been designed for nonholonomic mobile robot with stochastic disturbance by using the integrator backstepping procedure based on the state-scaling technique.There are several interesting problems of the controller design for the same stochastic nonholonomic mobile robot, for example, the tracking control and the adaptive control, and the further extensions to more general chained-form nonholonomic systems with stochastic disturbance.These directions are all under the current research.

1 4. 1 6
Mathematical Problems in Engineeringwhose infinite generator along the first equation of 1.1 satisfies

Figure 2 :
Figure 2: The Logic operation of switching.

Figure 3 :
Figure 3: The responses of closed-loop system with nonzero initial heading angle.

Figure 4 :
Figure 4: Responses of closed-loop system with zero initial heading angle by using switching 6.2 .

Figure 5 :
Figure 5: Responses of closed-loop system with zero initial heading angle by using switching 7.2 .
Theorem 4.2.Under assumptions (A1) and (A2), for every x 1 t 0 / 0 and any x 2 t 0 , x 3 t 0 , with an appropriate choice of the design parameters λ and c 1 , the closed-loop system consists of 3.1 , 3.2 , 4.3 , and 4.22 has a unique solution which is 4th moment exponential stable.