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

Authors: Hua Chen; Wen Chen; Chaoli Wang; Dongkai Zhang; Binwu Zhang
Journal: Journal of Function Spaces and Applications
 (2013)

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## Abstract

The finite-time stabilization problem of dynamic nonholonomic wheeled mobile robots with parameter uncertainties is considered for the first time. By the equivalent coordinate transformation of states, an uncertain 5-order chained form system can be obtained, based on which a discontinuous switching controller is proposed such that all the states of the robots can be stabilized to the origin equilibrium point within any given
settling time. The systematic strategy combines the theory of finite-time stability with a new switching control design method. Finally, the simulation result illustrates the effectiveness of the proposed controller.


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## Body

## 1. 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 [2–4], discontinuous feedback control laws [5–8], 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, finite-time stabilization problems have been studied mostly in the contexts of optimality, controllability, and deadbeat control for several decades [14–16]. 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 [19–21]. 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: Section2 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.


## 2. Problem Statement

A class of nonholonomic wheeled mobile robots are shown in Figure1, 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 r are identical for all the rear wheels, 2R is the length of the fixed two rear wheels, where r and R are known positive constants. Its kinematic and dynamics model can be described by the following differential equations [13]:
(1)x˙=vcosθ,y˙=vsinθ,θ˙=ω,mv˙=τ1+τ2r,Iω˙=Rr(τ1-τ2),
where x, y are the position of the mass center P of the robot moving in the plane and m, I are the mass and inertia of mobile robots; respectively, v is the forward velocity, ω is the steering velocity and θ denotes its heading angle from the horizontal axis, and τ1, τ2 are driving torques on the right and left rear wheels.Figure 1
Nonholonomic wheeled mobile robot.The geometric and inertia parameters(r,R,m,I) are all assumed to be unknown positive constants, but are bounded by some known positive constants (rm,rM,Rm,RM,mm,mM,Im,IM), that is,
(2)0<rm≤r≤rM,0<Rm≤R≤RM,0<mm≤m≤mM,0<Im≤I≤IM.
One of the equilibrium states of systems (1) is (x,y,θ,v,ω)=(0,0,0,0,0).The control objective is to design a discontinuous state feedback lawτ1(x,y,θ,v,ω), τ2(x,y,θ,v,ω) such that the state trajectory (x,y,θ,v,ω) of dynamic nonholonomic mobile robot system (1) starting from an arbitrary initial state (x(0),y(0),θ(0),v(0),ω(0)) converges to the origin equilibrium point in a finite time with the unknown parameters (r,R,m,I) satisfying (2).As pointed out in [11], take an orthogonal coordinate transformation
(3)(x1x2x3x4x5)=(00100cosθsinθ000sinθ-cosθ0000000100010)(xyθvω).
Then system (1) can be converted to the following equation:
(4)x˙1=x4,x˙2=x5-x3x4,x˙3=x2x4,x˙4=c1u1,x˙5=c2u2,
where u1=τ1-τ2, u2=τ1+τ2 are new control inputs and c1=R/rI, c2=1/rm are new unknown parameters with their bounds derived from (2) as follows:
(5)0<c1m=RmrMIM≤c1≤RMrmIm=c1M,0<c2m=1rMmM≤c2≤1rmmm=c2M.
Because the coordinate transformation (3) is globally invertible and does not change the origin, it is obvious that the equilibrium point (x1,x2,x3,x4,x5)=(0,0,0,0,0) is finite-time stable for its closed-loop system of (4) it is means that (x,y,θ,v,ω)=(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 finite-time 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 finite-time 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.The following definition and lemmas are needed for our controller design later.Definition 1 (see [14–16]).
Consider a time-invariant system in the form of(6)ξ˙=f(ξ),f(0)=0,ξ∈Rn,
where f:U^0→Rn is continuous on an open neighborhood U^0 of the origin. The equilibrium ξ=0 of the system is (locally) finite-time stable if (i) it is asymptotically stable, in U^, an open neighborhood of the origin, with U^⊆U^0; (ii) it is finite-time convergent in U^, that is, for any initial condition ξ0∈U^∖{0}, there is a settling time T>0 such that every solution ξ(t,ξ0) of system (6) is defined with ξ(t,ξ0)∈U^∖{0} for t∈[0,T) and satisfies limt→Tξ(t,ξ0)=0, and ξ(t,ξ0)≡0 if t≥T. Moreover, if U^=Rn, the origin ξ=0 is globally finite-time stable.Lemma 2 (see [15, 18]).
Suppose there exist aC1 positive-definite and proper function V-1:Rn→R, real numbers k->0, β-∈(0,1), such that V-˙1+k-V-1β- is negative semidefinite. Then, the origin is a globally finite-time stable equilibrium point of system (6).Lemma 3 (see [18]).
For the uncertain time-varying chained form system:(7)z˙1=q1z2u0,⋮z˙n-1=qn-1znu0,z˙n=qnu+ψn(z),
where z=[z1,z2,…,zn]T∈Rn, u∈R are the state and control input, respectively; qi>0  (i=1,2,…,n) are uncertain parameters located in known intervals, that is,
(8)0<qi∈[qimin,qimax],i=1,2,…,n;ψn(z) is an uncertain function satisfying
(9)|ψn(z)|≤M∑i=1n‍|zi|,M>0,
and u0(t) satisfies 0<γ≤|u0(t)|≤γ^.
Letri>0, βi-1>0, i=1,2,…,n and k=p/q-1<0 (with q>0 and p>0 odd integers) be real numbers satisfying
(10)r1=1,…,ri=ri-1+k,ri>-k>0,i=1,2,…,n,β0=r2,(βi+1)ri+1=(βi-1+1)ri>0,00000000000000000000i=1,2,…,n-1.
Then, finite-time stabilizing control law of (7) can be constructed in the form of
(11)u=vn=-γlnwn(rn+k)/(rnβn-1)-Mqnmin(∑i=1n‍|zi|)sgn(wn),
where wn is defined as follows:
(12)w1=z11+k,wi(z1,…,zi)=ziβi-1-vi-1βi-1,vi-1=-li-1wi-1(ri-1+k)/(ri-1βi-2)sgn(u0),i=2,…,n,
where sgn(·) denotes the sign function and li>0, i=1,…,n are suitable constants.Proof.
See [18] for details.


## 3. 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 {x1,x4}-subsystem
(13)x˙1=x4,x˙4=c1u1,
and the other describes the rest of (4), that is, {x3,x2,x5}-subsystem
(14)x˙3=x2x4,x˙2=x5-x3x4,x˙5=c2u2.
By designing u1, the state x4 of (13) can be driven to any predetermined point in a finite time, based on which (x3,x2,x5) of (14) can be stabilized to (0,0,0) by designing the finite-time controller u2, and the last step is to redesign u1 such that (x1,x4) can be driven to (0,0) in a finite time.The following are the switching controller design and the corresponding proofs of Theorems4 and 6.Theorem 4.
Givenc0>0, for system (4), take the following switching control law.
Step 1. Let u1=-k1sgn(x4-c0)|x4-c0|α1, u2=0, where k1>0, α1∈(0,1). Until x4=c0, then go to Step 2.
Step 2. Let
(15)u1=-k1sgn(x4-c0)|x4-c0|α1,u2=v~3=-l3w~3(r3+k)/(r3β3-1)-c0c2m(∑i=13‍|yi|)sgn(w~3),
where
(16)w~1=y11+k,w~i(y1,…,yi)=yiβi-1-v~i-1βi-1,v~i-1=-li-1w~i-1(ri-1+k)/(ri-1βi-2),i=2,3,y1=x3,y2=c0x2,y3=c0x5-c02x3.
Unitll yi=0, i=1,2,3,  then go to Step 3.
Step 3. Let u2=0,
(17)u1=v-2=-l2w-2(r2+k)/(r2β2-1),
where
(18)w-1=x11+k,w-2(x1,x4)=x4β1-v-1β1,v-1=-l1w-1(r1+k)/(r1β0).
Until x1=x4=0, stop.
Then system (4) can be stabilized to the origin equilibrium point in a finite time by the switching controller Step 1–Step 3.Proof.
In the first step, letη=x4-c0, V1=(1/2)η2, then
(19)η˙=x˙4=c1u1=-c1k1sgn(η)|η|α1,V˙1=ηη˙=-c1k1|η|1+α1=-c1k1(2V1)(1+α1)/2,
which means that V˙1+c1k1(2V1)(1+α1)/2≤0; according to the conclusion of Lemma 2, there exists a finite time T1>0 such that η=0 for all t≥T1, that is, x4=c0 after this time.
In Step 2, for the subsystem (14)
(20)x˙3=x2c0,x˙2=x5-x3c0,x˙5=c2u2.
Let
(21)(y1y2y3)=(1000c00-c0200)(x3x2x5).
One has
(22)y˙1=y2,y˙2=y3,y˙3=c0c2u2-c02y2.
Because det(A)=c02>0, where
(23)A=(1000c00-c020c0),
the finite-time stabilization problems of (20) and (22) are equivalent.
Setzi=yi  (i=1,2,3), q1=q2=u0=1, q3=c0c2, ψ3(z)=-c02z2, and from Lemma 3, system (20) can be stabilized to zero by u2 in Step 2 after some finite time T2; that is, x3=x2=x5=0 for all t≥T2.
Similarly, consider subsystem (13) again in the last step
(24)x˙1=x4,x˙4=c1u1.
Set z1=x1, z2=x4, q1=u0=1, q2=c1, ψ2(z)=0, by using Lemma 3, it is clear that there exists a finite time T3 such that system (13) can also be stabilized to zero after t≥T3.
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:
(25)x=x2cosθ+x3sinθ=x2cosx1+x3sinx1,y=x2sinθ-x3cosθ=x2sinx1-x3cosx1,θ=x1,v=x5,ω=x4.
Therefore, the finite-time switching controller for the original robot system (1) can be stated as follows.Theorem 6.
Givenc0>0, for system (1), take the following switching control law.
Step 1′. Let
(26)τ1=-12k1sgn(ω-c0)|ω-c0|α1,τ2=12k1sgn(ω-c0)|ω-c0|α1,
where k1>0, α1∈(0,1). Until ω=c0, then go to Step 2′.
Step 2′. Let
(27)τ1=-12((∑i=13‍|yi|)k1sgn(ω-c0)|ω-c0|α1+l3w~3(r3+k)/(r3β3-1)+c0c2m(∑i=13‍|yi|)sgn(w~3)),τ2=-12(l3w~3(r3+k)/(r3β3-1)+c0c2m(∑i=13‍|yi|)sgn(w~3)-k1sgn(ω-c0)|ω-c0|α1∑i=13‍),
where
(28)w~1=y11+k,w~i(y1,…,yi)=yiβi-1-v~i-1βi-1,v~i-1=-li-1w~i-1(ri-1+k)/(ri-1βi-2),i=2,3,y1=xsinθ-ycosθ,y2=c0(xcosθ+ysinθ),y3=c0v-c02(xsinθ-ycosθ).
Unitl yi=0, i=1,2,3, then go to Step 3′.
Step 3′. Let
(29)τ1=-12l2w-2(r2+k)/(r2β2-1),τ2=12l2w-2(r2+k)/(r2β2-1),
where
(30)w-1=θ1+k,w-2(θ,ω)=ωβ1-v-1β1,v-1=-l1w-1(r1+k)/(r1β0).
Until θ=ω=0, stop.
Then system (1) can be stabilized to the origin equilibrium point in a finite time by the switching controller Step 1′–Step 3′.Proof.
According to (25), we have the following:
(31)(xyθvω)=B(x1x2x3x4x5),
where
(32)B=(0cosx1sinx1000sinx1-cosx100100000000100010).
Because the determinant det(B)=1≠0, then it is obvious to see that (x,y,θ,v,ω)→0 is equivalent to (x1,x2,x3,x4,x5)→0.
On the other hand, the switching controller(u1,u2) in Theorem 4 can be used to stabilize the states (x1,x2,x3,x4,x5) of system (4) in a finite time; hence, the control task is changed to find the relation between the original controller (τ1,τ2) of system (1) and the controller (u1,u2) of system (4).
Comparing system (4) with system (1), we have the following:
(33)τ1=u2+u12,τ2=u2-u12,c1=RrI,c2=1rm,
from which, and by using the switching controller of Step 1–Step 3 in Theorem 4, we can solve the corresponding (τ1,τ2) and thus the conclusion can be obtained in Theorem 6.
This completes the proof of the theorem.


## 4. Simulations

In this section, the discontinuous switching controller proposed in theorems above is used to show how to stabilize the state(x1,x2,x3,x4,x5) of (4) and the state (x,y,θ,v,ω) of (1) to the zero equilibrium point in a finite time. We will demonstrate the effectiveness of our methods by a numerical example.In the following simulation, we assume thatrm=0.5, rM=2.0, Rm=1.5, RM=3.5, mm=0.6, mM=1.8, Im=1.2, IM=3.0. The initial condition (x(0),y(0),θ(0),v(0),ω(0)) of system (1) is (-0.2,-1.0,0.8,0.5,1.5). From (3) and (5), the initial value (x1(0),x2(0),x3(0),x4(0),x5(0)) of system (4) is (0.8,-0.86,-0.84,1.5,0.5), and c1m=0.25, c1M=5.83, c2m=0.28, c2M=3.33. Given c0=1.0, the design parameters are taken as follows: k1=2.5, α1=0.6, k=p/q-1=7/9-1=-2/9, r1=1, r2=7/9, r3=5/9, β0=7/9, β1=9/7, β2=11/5, l1=1.2, l2=1.5, l3=2.8. The settling time in every step is given in advance; T1=5s, T2=20s, T3=30s.Figures2–5 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 T3=30s. Observing Figure 2, in time interval 0~5 s, the first step control task is completed, that is, x4=c0=1 as t≥5s. Next, from Figure 3, it is clear that (x3,x2,x5) can be stabilized to zero by the controller u2 in Step 2 as t≥20s and remain unchanged. Finally, the controller u1 in Step 3 drives (x1,x4) to zero in the settling time t≤T3.Figure 2
State variablesx1, x4.Figure 3
State variablesx3, x2, and x5.Figure 4
Original state variablesθ, ω.Figure 5
Original state variablesx, y, and v.Figures4 and 5 show the finite-time convergence of the state variables (x,y,θ,v,ω) of the original robot system (1) exactly as the state of the transformed system.


## 5. 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.



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