fixed-time stabilization for a wheeled mobile robot with

Fixed-Time Stabilization for A Wheeled MobileRobot With Actuator Dead-Zones

Zicheng Yang, Yulu Zhao and Fangzheng Gao

Abstract—In this paper, the problem of fixed-time stabiliza-tion is addressed for a unicycle-type wheeled mobile robotwith actuator dead-zones. A novel switching control strategyis given to overcome the obstacle that the presence of actuatordead-zones renders the traditional feedback control techniqueinapplicable to such mobile robot. Then, by employing theadding a power integrator(API) technique, a state feedbackcontroller is successfully developed to regulate all states ofclosed-loop system (CLS) to zero in a given fixed time. Finally,simulation results are given to confirm the efficacy of theproposed method.

Index Terms—wheeled mobile robot, actuator dead-zones,adding a power integrator (API), fixed-time stabilization.

I. INTRODUCTION

THE wheeled mobile robot (WMR) has attracted muchattention in the past years because it wide application-

s in entertainment, security, war, rescue missions, spacialmissions, assistant health-care, etc [1-3]. An important fea-ture of WMR that the number of control inputs is fewerthan the number of freedom degrees, leads to the controlof WMR challenging. As pointed out by Brockett in [4],there is not any continuous time-invariant state feedbackto stabilize such category of nonlinear systems. To addressthis difficulty, a number of control approaches have beenproposed, mainly including time-varying feedback [5-7] anddiscontinuous time-invariant feedback [8,9]. Thanks to thesevalid approaches, many significant results have been made,e.g., [10-16] and the references therein.

Noted that majority existing results mainly centre aroundthe asymptotic behavior of system trajectories as time vergesto infinity. However, in practical applications, the CLS isdesired to have the property that trajectories converge to theequilibrium in finite time. Moreover finite-time stable systempossesses the superior properties of fast response, goodrobustness and disturbance rejection [17]. Motivated by thesefacts, the research on finite-time control has become popularrecently [18-20]. Especially, as the preliminary research, thework [21] addressed the problem of finite-time stabilizationby state feedback for a family of nonholonomic systemswith some weak drifts. Whereafter, the problems of adaptive

This work was partially supported by the National Natural ScienceFoundation of China under Grant 61873120, the National Natural Sci-ence Foundation of Jiangsu Province under Grant BK20201469, the OpenResearch Fund of Jiangsu Collaborative Innovation Center for Smart Dis-tribution Network, Nanjing Institute of Technology Grants XTCX201909,XTCX202006 and the Qing Lan project of Jiangsu Province.

Zicheng Yang is an undergraduate in the School of Automa-tion, Nanjing Institute of Technology, Nanjing 211167, P. R. [email protected]

Yulu Zhao is an undergraduate in the School of Automation,Nanjing Institute of Technology, Nanjing 211167, P. R. [email protected]

Fangzheng Gao is an associate professor in the School of Automa-tion, Nanjing Institute of Technology, Nanjing 211167, P. R. [email protected]

finite-time stabilization with linear/nonlinear parameteriza-tion were studied in [22] and [23]. By relaxing the limitationon system growth, the authors in [24] extended the workof [21] to a general category of nonholonomic systems.An output feedback controller was developed in [25] tofinite-time stabilize a category of nonholonomic systems infeedforward-like form. Later, this result is further extended tothe high order case in [26] and [27]. But a common drawbackof the aforementioned studies is that the convergence timedepends on initial system conditions, which renders thatthe desired performance in an exact preset time cannotbe achieved. Recently, to remove the limitation of finite-time algorithm, a novel finite-time stability concept thatrequires the convergence time of a global finite-time stablesystem being bounded independent of initial conditions, wasintroduced in [28]. Such stability, usually called fixed-timestability, offers a new perspective to study the finite-timecontrol problems and has stimulated some interesting results[29-32]. However, the effect of the actuator dead-zone isignored in the aforementioned results.

In reality, owing to physical limitations of device, in-put dead-zone nonlinearity inevitably are suffered duringoperation in many real systems. Such unexpected propertycould seriously degrade the system’s performance [33-35].Therefore, the interesting question naturally arises: For aWMR with actuator dead-zones, is it possible to devise acontroller to achieve the fixed-time stabilization? If possible,how can one design it?

Motivated by the above observations, this paper focuses onsolving the problem of fixed-time stabilization of nonholo-nomic WMR with actuator dead-zones. The contributionsare highlighted as follows. (i) The fixed-time stabilizationproblem of nonholonomic WMR with actuator dead-zonesis studied. (ii) A novel switching control strategy is givento overcome the obstacle that the presence of actuator dead-zones renders the traditional discontinuous feedback controltechnique inapplicable to nonholonomic systems. (iii) Byemploying the API technique, a systematic state feedbackcontrol design procedure is proposed to ensure all states ofthe CLS to zero for any given fixed time.

Notations. In this paper, the notations used are fairlystandard. Specifically, for any c > 0 and η ∈ R, the function[η]c is defined as [η]c = sign(η)|η|c with the standard signumfunction sign(·).

II. PROBLEM STATEMENT AND PRELIMINARIES

Consider a tricycle-type WMR shown in Fig.1. The kine-matic equations of this robot are represented by

xc = v cos θ,yc = v sin θ,

θ = ω,(1)

IAENG International Journal of Computer Science, 48:3, IJCS_48_3_07

Volume 48, Issue 3: September 2021

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Fig. 1. The planar graph of a mobile robot.

where (xc, yc) denotes the position of the center of mass ofthe robot, θ is the heading angle of the robot, v is the forwardvelocity while ω is the angular velocity of the robot.

Introducing x0 = θ,x1 = xc sin θ − yc cos θ,x2 = xc cos θ + yc sin θ,u0 = w,u1 = v,

(2)

and taking into the inevitably presence of the actuator dead-zones in reality, the dynamics of (1) can be modelled as x0 = D0(u0),

x1 = x2D0(u0),x2 = D1(u1)− x1D0(u0),

(3)

where Dj (j = 0, 1) is the dead-zone input nonlinearitiesdescribed by

Dj(uj) =

mj(uj − bj), uj ≥ bj ,0, −bj < uj < bj ,

mj(uj + bj), uj ≤ −bj ,(4)

with mj and bj being the slopes and the breakpoints of thedead-zone characteristic, respectively.

Remark 1. Consider the system (3) with free of actuatordead-zones. With the traditional discontinuous feedback con-trol technique, one can design u0 = −x0 to ensure x0(t) = 0(or equivalently, u0(t) = 0) for all t ≥ 0 provided x0(0) = 0.Then, by introducing the discontinuous change of coordinatesz1 = x1/u0 and z2 = x2 to overcome the obstacle that thex-subsystem is uncontrollable in the case of u0 = 0, one canchange the x-subsystem into a strict-feedback-like form{

z1 = z2 + z1,z2 = u1 − x2

0z1,(5)

and solve the stabilization problem by the well-known‘backstepping’ technique or its variations. However, whenthe actuator dead-zones are involved, it is clear that suchtransformation fails to work. That is, the traditional discontin-uous feedback control technique is inapplicable to the dead-zone constrained nonholonomic systems. Consequently, newcontrol techniques are wanted for solving the problem ofglobal stabilization of the dead-zone constrained system (3).

The following, assumption, definitions and lemmas willserve as the basis of the coming control design and perfor-mance analysis.

Assumption 1. There are positive constants m, m, b and bsuch that for j = 0, 1, one has m ≤ mj ≤ m and b ≤ bj ≤ b.

Definition 1[17]. Consider the nonlinear system

x = f(t, x) with f(t, 0) = 0, x ∈ Rn, (6)

where f : R+ × Rn → Rn is continuous with respect to x.The equilibrium x = 0 of the system is globally finite-timestable if it is Lyapunov stable and for any initial conditionx(t0) ∈ Rn at any given initial time t0 ≥ 0, there is asettling time T > 0 , such that every x(t, t0, x(t0)) of system(6) satisfies limt→T x(t, t0, x(t0)) = 0 for t ∈ [t0, T ) andx(t, t0, x(t0)) = 0 for any t ≥ T .

Lemma 1[17]. Consider the nonlinear system described in(6). Suppose there is a C1 function V (t, x) defined on Rn,class K functions π1 and π2, real numbers c > 0 and 0 <α < 1, for t ∈ [t0, T ) and x ∈ Rn such that

π1(|x|) ≤ V (t, x) ≤ π2(|x|), ∀t ≥ t0,∀x ∈ Rn,

and

V (t, x) + cV α(t, x) ≤ 0, ∀t ≥ t0 , ∀x ∈ Rn.

Then, the origin of (6) is globally finite-time stable with

T ≤ V 1−α(t0, x(t0))

c(1− α).

Definition 2[28]. The origin of system (6) is referred to beglobally fixed-time stable if it is globally finite-time stableand the settling time function T (x0) is bounded, that is, thereexists a positive constant Tmax such that T (x0) ≤ Tmax,∀x0 ∈ Rn.

Lemma 2[28]. Consider the nonlinear system (6). Supposethere exist a C1, positive definite and radially unboundedfunction V (x) : Rn → R and real numbers c > 0, d > 0,0 < α < 1, γ > 1, such that

V (x) ≤ −cV α(x)− dV γ(x), ∀x ∈ Rn.

Then, the origin of system (6) is globally fixed-time stableand the settling time T (x0) satisfies

T (x0) ≤ Tmax :=1

c(1− α)+

1

d(γ − 1), ∀x0 ∈ Rn.

Lemma 3[36]. For any x, y ∈ R, and a constant a ≥ 1,one has

|x+ y|a ≤ 2a−1|xa + ya|;

(|x|+ |y|)1/a ≤ |x|1/q + |y|1/a ≤ 2(a−1)/a(|x|+ |y|)1/a.

Lemma 4[36]. If c, d are positive constants, then for anyreal-valued function δ(u, v) > 0, one has

|u|c|v|d ≤ c

c+ dδ(u, v)|u|c+d +

d

c+ dδ−c/d(u, v)|v|c+d.

III. FIXED-TIME CONTROL

In this section,a constructive procedure for the finite-timestabilizer design of system (3) is given for any given settlingtime T > 0.



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A. Fixed-time stabilization of the x-subsystem

For the x0-subsystem, take

u0 = u∗0, (7)

where u∗0 is a constant satisfying u∗

0 > b. Then, the x-subsystem in this case is described as{

x1 = h1x2,x2 = h2u1 +Φ2(x2),

(8)

with h1 = D0(c∗0), h2 = 1 and Φ2 = −x1D0(c

∗0). As a

result, it is easily checked from Assumption 1 thatProposition 1. Under (7), the solution of the x0-

subsystem x0(t) is well-defined on [0,+∞) and there arepositive constants C, hi1 and hi2, i = 1, 2 such thathi1 ≤ hi ≤ hi2 and |Φ2| ≤ C|x1|.

Next, the system (8) will be stabilized within the settlingtime θT by employing API technique. Before proceeding,we take r1 = 1 and ri+1 = ri + τ > 0, i = 1, 2, 3 withτ ∈ (− 1

n , 0) being a negative number, and introduce thefollowing coordinate transformation:

ξi = [xi]1ri − [αi−1]

1ri ,

αi = −gri+1

i (xi)[ξi]ri+1 , i = 1, 2,

(9)

where α0 = 0, α2 = u1 and gi(xi) > 0 is a C1 function tobe specified later.

We further define Wi : Ri → R as follows:

Wi(xi) =

∫ xi

αi−1

[[s]

1ri − [αi−1]

1ri

]2−ri+1

ds. (10)

The detailed design procedure is elaborated as follows.Step 1. For the x1-subsystem of (8), take x2 as the virtual

control input. Choose V1 = W1 and g1 = ((1 + l1 +

l2|ξ1|p)/h11)1r2 with design parameters l1 > 0, l2 > 0 and

p > −τ to be determined later, one has

V1 ≤ −(1 + l1)|ξ1|2 − l2|ξ1|2+p + h1[ξ1]2−r2(x2 − α1).

(11)Step 2 . Consider the second Lyapunov function V2 =

V1 +W2. It can be deduced from (15) that

V2 ≤ −(1 + l1)|ξ1|2 − l2|ξ1|2+p

+h1[ξ1]2−r2(z2 − α1) + [ξ2]

2−r3Φ2

+h2[ξ2]2−r3D1(u1) +

∂W2

∂z1h1z2.

(12)

First, we observe from Lemmas 3 and 4 that

h1[ξ1]2−r2(z2 − α1) ≤ 2h1|ξ1|2−r2 |ξ2|r2

≤ 1

2|ξ1|2 + φ21|ξ2|2,

(13)

where φ21 ≥ 0 is a C1 function.Then, by using Lemmas 3 and 4, we have

[ξ2]2−r3Φ2 +

∂W2

∂z1h1z2 ≤ 1

2|ξ1|2 + φ22|ξ2|2, (14)

where φ22 ≥ 0 is a C1 function.Choosing

g2 =( l1 + φ21 + φ22 + l2|ξ2|p

h21

) 1r3, (15)

and substituting (13), (14 and (15) into (12), we have

V2 ≤ −l1

2∑j=1

|ξj |2 − l2

2∑j=1

|ξj |2+p

+h2[ξ2]2−r3 (D1(u1)− α2) .

(16)

Thus, from Assumption 2, the control u1 is designed as

u1 =

α2

m+ b, α2 > 0,

0, α2 = 0,α2

m− b, α2 < 0,

(17)

which rendersD1(u1)− α2

=

m1

(ζ∗n+1

m+ b− b1

)− α2, α2 > 0,

0, α2 = 0,

m1

(ζ∗n+1

m− b+ b1

)+ (−α2), α2 < 0.

=

1

m(m1 −m)α2 +m1(b− b1) > 0, α2 > 0,

0, α2 = 0,1

m(m1 −m)α2 −m1(b− b1) < 0, α2 < 0.

(18)By noting −[ξ2]

2−r3α2 ≥ 0, one getssuch that

V2 ≤ −l1

2∑j=1

|ξj |2 − l2

2∑j=1

|ξj |2+p, (19)

where V2 =∑2

j=1 Wj .Consequently, the following result is obtained.

Proposition 2. If the controller u1 of system (8) isspecified by (17) with design parameters l1 > 0, l2 > 0and p > −τ satisfying

2(τ − 2)

θl1τ+

(2− τ)22+2p+τ

2−τ

θl2(p+ τ)< T, (20)

then the equilibrium x = 0 of CLS is globally fixed-timestable and all the trajectories converge to zero before a fixedtime θT .

Proof. According to (xi − αi−1)([zi]1ri − [αi−1]

1ri ) ≥ 0,

we easily verify that V2 =∑2

j=1 Wj is positive definiteand radially unbounded. Moreover, we have the followingestimation for V2.

V2 =

2∑j=1

Wj ≤ 2

2∑j=1

|ξj |2−τ . (21)

Letting α = 2/(2− τ), it is not difficult to obtain that

−2∑

j=1

|ξj |2 ≤ −1

2V α2 . (22)

On the other hand, taking (21) into account, it can bededuced that

−2∑

j=1

|ξj |2+p = −2∑

j=1

(|ξj |2−τ

) 2+p2−τ

≤ −21−2+p2−τ

( n∑j=1

|ξj |2−τ) 2+p

2−τ

≤ −2−γ21−γV γ2 ,

(23)



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where γ = (2 + p)/(2− τ).Therefore, by considering (19), (22) and (23), it follows

thatV2 ≤ −1

2l1V

α2 − l22

−γ21−γV γ2 . (24)

Since α < 1 and γ > 1, from Lemma 2, we conclude thatthe equilibrium z = 0 of the closed-loop system is globallyfixed-time stable and the settling time function T1 satisfies

T1 ≤ 2

l1(1− α)+

2γnγ−1

l2(γ − 1)

=2(τ − 2)

l1τ+

(2− τ)22+p2−τ n

p+τ2−τ

l2(p+ τ)< θT.

(25)

0 1 2 3 4 5 6 7 8 9 10−0.5

0

0.5

1

1.5

2

2.5

3

Time(sec)

x0

Fig. 2. x0.

B. Fixed-time stabilization of the x0-subsystem

From Proposition 2, it si known that x(t) ≡ 0 when t ≥θT . Since x(t) = 0, x(t) = 0 holds for t ≥ θT in spitethat a new controller will be designed for u0 when t ≥ θT .Hence, it is just needed to stabilize the x0-subsystem in afixed time θT . In this case, for the x0-subsystem, the controlα0 is taken as

α0 = −(m0 +m1|x0|q)[x0]σ, (26)

where

u0 =

α0

m+ b, α0 > 0,

0, α0 = 0,α0

m− b, α0 < 0,

(27)

and 0 < σ < 1, m0 > 0, m1 > 0 and q > 1 − σ are thedesign parameters to be determined later.

Proposition 3. If design parameters 0 < σ < 1, m0 > 0,m1 > 0 and q > 1− σ in (27) satisfy

2

m0(1− σ)(1− θ)+

2

m1(σ + q − 1)(1− θ)< T, (28)

then the state x0 is regulated to zero within a fixed settlingtime (1− θ)T .

Proof. The proof of Proposition 3 follows the same lineof the proofs of Proposition 2.

Consequently, the following theorem is obtained to sum-marize the main result of the paper.

Theorem 1. If the following switching control strategywith the appropriate chosen design parameters is applied tosystem (3), then the states of the CLS are regulated to zerowithin any given settling time T .

0 1 2 3 4 5 6 7 8 9 10−1

−0.5

0

0.5

1

1.5

Time(sec)

x1

x2

Fig. 3. x1 and x2.

IV. SIMULATION RESULTS

In this section, we illustrate the effectiveness of the pro-posed approach by taking the dead-zone input nonlinearitiesDj , j = 0, 1 described by (4) with mj = 1 + 0.2 sin t,bj = 0.3 + 0.1 cos t respectively. In this situation, it is clearthat Assumptions 1 holds with m = 0.8, m = 1.2, b = 0.2and b = 0.4. Furthermore, by setting u0 = c∗0 with c∗0 > bbeing a positive constant, it is easily verified that Assumption2.1 is satisfied with ν = 0, λ1 = λ2 = λ3 = 1 andφ2 = m(c∗0 − b).

Taking τ = −1/3 and following the design proceduregiven in Section III, a state feedback controller of from (17)is constructed such that the states of the x-subsystem of (3)are globally regulated to zero within a fixed settling time θT .

Then, when t ≥ θT , for the x0-subsystem, switch thecontrol input u0 to (26) such that the state x0 is regulated tozero within a fixed settling time (1− θ)T .

In the simulation, by choosing the fixed time T = 10and the gains for the control laws as u∗

0 = 1, l1 = 4,l2 = 5, p = 2, θ = 0.8, q1 = q2 = 1, p = 0.8,p = 1.2, σ = 0.5 and m0 = m1 = q = 2, Figs. 2 and3 is obtained to exhibit the responses of the closed-loopsystem with (x0(0), x1(0), x2(0)) = (−0.5,−1, 1). Fromthe figures, it can be seen that the states of the closed-loop switched system converge to zero in a given fixed time,which demonstrates the effectiveness of the control methodproposed in this paper.

V. CONCLUSION

This paper has studied the problem of fixed-time stabiliza-tion by state feedback for nonholonomic WMR with actuatordead-zones. By employing the API technique, a constructivestate feedback design procedure is given, which together witha novel switching control strategy, ensures that the states ofthe CLS are regulated to zero for any given fixed time.



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fixed-time stabilization for a wheeled mobile robot with

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