Finite-time trajectory tracking control for rigid 3-DOF ... · results of a 3-DOF manipulator shows that trajectory tracking was achieved with a ﬁnite period of settling time. The

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Finite-time trajectory tracking control forrigid 3-DOF manipulators withdisturbancesZIYANG CHEN1, XIAOHUI YANG1,XIAOLONG ZHANG1 AND PETER X. LIU .1,2 (SeniorMember, IEEE)1College of Information Engineering, Nanchang University, Nanchang 330031, Jiangxi, China2Department of Systems and Computer Engineering, Carleton University, Ottawa, ON Canada K1S 5B6

Corresponding author:[email protected]

This work was supported in part by the National Natural Science Foundation of China (51765042, 61463031, 61773051), JiangxiProvincial Department of Science and Technology (JXYJG-2017-02), Jiangxi Natural Science Foundation (20171ACB20007) and JiangxiProvincial Department of Science and Technology (20121BBE50023, 20133BCB22002).

ABSTRACT This paper addresses the robust finite time trajectory tracking control problem for a rigidthree-degrees-of-freedom (3-DOF) manipulator system, which is subject to model uncertainty and externaldisturbances. The disturbances are assumed to be upper bounded. The proposed method incorporates afinite time velocity observer and disturbance observer into the proposed nonlinear control scheme and theconditions of global finite time stability are established. The global finite-time stability of the controlledsystem is proved by using the finite-time Lyapunov stability theory. Simulation results of a 3-DOFmanipulator show that the trajectory tracking was achieved with a finite period of settling time.

INDEX TERMS Finite-time tracking, global finite-time stability, disturbance observer, velocity observer.

I. INTRODUCTION

THE problem of high-precision, fast-response trajectorytracking control of robotic manipulators has long been

a research hotspot [1]–[4]. A number of strategies have beenproposed for this purpose, including computed torque control[5]–[8] and inverse-dynamics control [6], [9], to guaranteethe asymptotic stability of robotic manipulators system [10].However, the actual trajectories generated using these controlmethods usually cannot converge to the desired ones within afinite period of settling time. To address this issue, finite-timestabilization strategies have been developed, enabling roboticmanipulators to converge fast with high precision within afinite period of time [11]–[18].

Most finite-time control methods need joint position andvelocity measurements [15]–[20], which not only requireextra sensors, but also add to the cost, weight, size, and noiseto the servo systems of manipulators. A number of nonlinearvelocity observers have been proposed [21]–[23], however,most of these velocity observers provide estimation in thesense of infinite time. In order to provide faster velocity es-timation and higher estimation accuracy, finite time velocityobservers have been proposed [22].

In recent years, the disturbance observer (DO) has beenintroduced for robotic manipulator control [24], [25] in orderto deal with the robust problem associated with uncertainties,external disturbances, and unmodeled friction forces in robot

manipulator systems [3], [8], [9], [26]. In general, the mainpurpose of using DO is to further deduce external unknown oruncertain disturbance torques without the use of an additionalsensor [27], [28]. In particular, finite-time DO has beenproposed to ensure the finite-time convergence of disturbanceestimation [15].

Recently, Bouakrif [23] proposes a trajectory trackingstrategy with a nonlinear disturbance observer and velocityobserver for robot manipulators. Using position measure-ments only, it demonstrated that the actual trajectories rapidlytracked the desired trajectories under the conditions of modeluncertainties and external disturbances, however, the trajec-tory tracking was achieved asymptotically.

This paper introduces a finite-time nonlinear controlscheme for rigid 3-DOF manipulators with external dis-turbance and model uncertainty. The proposed method in-corporates a finite time velocity observer and disturbanceobserver into the proposed nonlinear control scheme and theconditions of global finite time stability are established. Theglobal finite-time stability of the controlled system is provedby using the finite-time Lyapunov stability theory. Simulationresults of a 3-DOF manipulator shows that trajectory trackingwas achieved with a finite period of settling time.

The rest of this paper is organized as follows. The dynam-ics of a rigid 3-DOF manipulator is introduced in Section 2.Section 3 describes the design of a finite-time disturbance

1



observer, and the input torque is generated by incorporatingthe disturbance observer with a nonlinear controller. Then,the finite-time velocity observer is introduced. In Section4, global finite-time stabilization of a 3-DOF manipulatorssystem is proved by using Lyapunov stability theory. InSection 5, numerical simulation are provided to verify thecorrectness of the theoretical derivation. Section 6 draws aconclusion and gives some suggestions for further research.

II. THE DYNAMICS OF A RIGID 3-DOF MANIPULATORIn this section, a rigid 3-DOF manipulator system, which suf-fers external disturbance and model uncertainty is presented.Later, according to the manipulator model, the state equationof the system is derived and the tracking error dynamics ofthe manipulator is obtained.

Consider a rigid 3-DOF manipulator system given by thefollowing model:

τ = M (θ) θ+B (θ)[θ · θ

]+C (θ)

[θ2]

+G (θ) + τd (1)

where θ = [θ1, θ2, θ3]T, θ, θ ∈ R3×1 are the position,

velocity, and acceleration vectors of manipulator joints, re-spectively. M (θ) ∈ R3×3 is the positive definite inertiamatrix. B (θ) ∈ R3×3 is the vector of Coriolis torques.C (θ) ∈ R3×3 represents the vector of centripetal forces.[θ · θ

]∈ R3×1 and

[θ2]∈ R3×1 are defined as

[θ · θ

]=[

θ1θ2, θ1θ3, θ2θ3

]Tand

[θ2]

=[θ1

2, θ2

2, θ3

2]T

. G (θ) ∈R3×1 is the vector of gravitational torques. τ = [τ1, τ2, τ3]

T

represents the vector of input torques. τd, which is (byassumption) absolutely continuous, represents the externaldisturbance and model uncertainty, and its derivative τd isassumed to be a locally bounded Lebesgue measurable toensure the existence of control τ . In consequence, ‖ τd ‖and ‖ τd ‖ are assumed to be upper estimated as follows:

‖ τd ‖≤ β0 (t) , ‖ τd ‖≤ β1 (t) (2)

where β0 (t) and β1 (t) are known non-negative functions.The rigid 3-DOF manipulator model is shown in Fig. 1.

FIGURE 1. Structure of 3–DOF manipulator

As seen from this figure, the first joint is the translationaljoint, which can move up and down. Both the second and thethird joints can rotate freely. The weights of the three jointsare m1 = 1kg, m2 = 1kg , and m3 = 1kg, respectively. Themovement range d of the first link, satisfies d = 0 − 0.21m.The lengths of the other two links are l2 = 0.2m and l3 =0.2m, respectively.

Therefore, according to the 3-DOF manipulator system in(1), matrices M , B, C, and G of this 3-DOF manipulator aregiven as follows:

M (θ) =

m11 m12 m13

m21 m22 m23

m31 m32 m33

, B (θ) =

00 000−l2l3m3 sin θ00 0

,C (θ) =

0 0 0

0 0 −1

2l2l3m3 sin θ

0 −1

2l2l3m3 sin θ 0

,G (θ) =

[(m1 +m2 +m3) g

00

].

(3)

The components of M are given by

m11 = m1 +m2 +m3

m12 = m13 = m21 = m31 = 0

m22 =1

3m2l

22 +

1

3m3l

23 +m3l

22 + l2l3m3 cos θ3

m23 =1

2l2l3m3 cos θ3 +

1

3m3l

23

m32 =1

2l2l3m3 cos θ3 +

1

3m3l

23

m33 =1

3m3l

23.

(4)The state variables x and v represent positions and veloci-

ties, respectively: {x = θ

v = x(5)

where x = [x1, x2, x3]T and v = [v1, v2, v3]

T . Thus, thestate equation of the robot manipulator can be described by{

x = v

v = f (x, v) + φ (x, v, t) + p (x) (τ +4u)(6)

where p (x)=M−1(x),f=−p (x)(B (x)〈v, v〉+C (x) v2

)−

p (x)G (x),φ = −p (x) τd, and4u = u− τ . u is the controlvector of the robot system.

Considering the saturation constraint of the joints’ actu-ators, we design the saturation function to limit the controlvector u:

u =

uupper if τ > uupper

τ if − ulower ≤ τ ≤ uupper−ulower if τ < −ulower

(7)

where uupper and ulower are the known values of the satura-tion function. Note that uupper 6= ulower is always satisfied.

Despite the uncertainties, external disturbances, un-modeled friction forces, and actuators’ saturation, all trajec-tories of manipulator joints can reach the desired trajectoriesin finite time for suitable values of the input torque τ .

2



xd = [xd1 , xd2 , xd3 ]T represents the desired trajectories and

vd = [vd1 , vd2 , vd3 ]T gives the derivatives of the desired

trajectories; thus, tracking errors can be described by ex =x − xd and ev = v − vd . Based on the state space model in(6) the tracking error dynamics of the manipulator are givenas follows:{

ex=ev

ev=f(x, v)+φ (x, v, t)+p (x) (τ +4u)−vd(8)

III. DESIGN OF THE CLOSED-LOOP CONTROL SYSTEMIn this section, we describe the design of the closed-loopcontrol system satisfying the conditions of global finite timestability based on the dynamics of the 3-DOF manipulatorgiven in the last section. Firstly we propose a disturbanceobserver to estimate the disturbance within finite time. Sec-ond, by incorporating the nonlinear sliding mode controllerwith the disturbance observer, input torque is determined toguarantee that the trajectory error converges to zero withinfinite settle time. In the following subsection, a finite timevelocity observer is designed to replace the velocity in thedisturbance observer and input torque with its estimation.Finally, the control input is acquired in virtue of modifiedinput torque and the saturation function, which indicates thatthe closed-loop control system is established.

A. DESIGN OF THE DISTURBANCE OBSERVER ANDINPUT TORQUEIn order to estimate the disturbance within finite settle time,the disturbance observer is designed as follows:

φ=−Kω − f (x, v)− γsigc (ω)

−β0 (t)λmax

(M−1

)sgn(ω)−|f(x, v)|sgn(ω)

z=−Kω+pu−vd−β0(t)λmax

(M−1

)sgn (ω)

−γsigc (ω)− | f (x, v) | sgn (ω)

τd=p−1φ, ω=z − ev

(9)

where φ and τd are the estimates of φ and τd, respectively.ω = [ω1, ω2, ω3]

T represents the subsidiary vector. Kω =[K1ω1,K2ω2,K3ω3]

T , γ = [γ1, γ2, γ3]T , and sigc (ω) =

[| ω1 |c sgn (ω1) , | ω2 |c sgn (ω2) , | ω3 |c sgn (ω3)]T , where

kj > 0, γj > 0 and 0 < c < 1 are arbitrary coeffi-cients, and sgn (ω) denotes the signum function of vectorω. λmax

(M−1

)is the maximum eigenvalue of M−1. Note

that variables x and v are assumed to be directly measurablehere.

According to the disturbance observer, τd can exactlyconverge to τd after finite time Ta, satisfying the followinginequality :

Ta ≤

ln

α

√√√√( 3∑j=1

s2j (0) + ω2j (0)

)1−c

+ δ

− ln δ

α (1− c)(10)

where parameters α and δ satisfy α = min (min εj ,min kj)and δ = min (min γj ,min ςj) in which εj and ςj are arbitrarycoefficient.

Then, for the objective of designing the input torque, anonlinear sliding mode controller is also introduced. In thispaper, we choose the sliding surface s = [s1, s2, s3]

T asfollows:

s = ω + ev + sig$x (ev) + sig$v (ψ (ex, ev)) . (11)

In (11), parameters $x = [$x1, $x2, $x3]T and $v =

[$v1, $v2, $v3]T are defined, where 0 < $xj < 1

is an arbitrary coefficient and $vj = $xj (2−$xj)−1

for j = 1, 2, 3. Also, parameter ψ (ex, ev) is given byψ (ex, ev) = [ψj (ex1, ev1) , ψj (ex2, ev2) , ψj (ex3, ev3)]

T ,where ψj (exj , evj)=(2−$xj)

−1 |evj |2−$xj sgn(evj)+exjfor j = 1, 2, 3. By incorporating the controller with adisturbance observer, the input torque τ can be chosen suchthat the control system achieves global finite-time stability.For this purpose, the control law of τ is proposed as follows:{

τ=p−1(−f(x, v)−sig$x(ev)−sig$v(ψ (ex, ev))+vd+τs)−∆u

τs=−εs−ςsigc (s)−φ−(β0 (t)λmax

(M−1

)+β2 (t)

)sgn(s)

(12)where function β2 (t) = β0 (t) ‖ ˙p (x)‖+β1 (t)λmax

(M−1

),

εs = [ε1s1, ε2s2, ε3s3]T , and ς = [ς1, ς2, ς3]

T are as defined.When the input torque τ is applied to the error dynamics ofthe manipulator in (8), the tracking error is able to reach thesliding surface s = 0 in finite time Ta, and then convergesto zero after finite time Tb. In other words, all tracking errorsreach zero within finite time Ttol = Ta + Tb.

B. DESIGN OF THE VELOCITY OBSERVER ANDCONTROL INPUTOwing to the drawbacks of direct measurement of jointvelocity, the method used in this work only measures positionx directly, while velocity v is estimated by velocity observerin adjustable finite time. Assume that velocity v is boundedas follows:

‖ v ‖≤ κ (13)

where κ is a known constant.Hence, the velocity observer is designed as follows:

˙x = v − (σ0 + σ3) sig (x− x)

−h (x, v) sgn (x− x)− σ2sigξ (x− x)˙v = −σ1v − σ3 | x− x | +f (x, v) + p (x)u

h (x, v) = (‖ v ‖ +κ)L1 (x, v) + L2 (x, v)

+σ1κ+σ2 (‖v‖+κ)ξ+λmax

(M−1

)β0(t)

(14)

where x = [x1, x2, x3]T and v = [v1, v2, v3]

T are estimatesof x and v, respectively. σ0, σ2, σ3 > 0, and 0 < ξ < 1are arbitrary parameters which can be set by users. Defineh(x, v)=B(x) 〈v, v〉+C(x)v2,L1(x, v)=λmax(M−1)(‖B‖+‖C‖)(‖v‖ + κ), L2(x, v)=2λmax(M−1)(‖B‖ + ‖C‖)‖v‖2,and σ1=1+σ0. Clearly, through this velocity observer, theobtained estimate of velocity v can converge to v within finitetime Tc, which satisfies the following inequality:

Tc ≤ln(σ0 (‖ex (0)‖+‖ev (0)‖)1−ξ+σ2

)−lnσ2

σ0 (1− ξ)(15)

3



where ex = x− x, ev = v − v.Remark 1: By substituting v for v, the disturbance observer

in (9) and input torque in (12) are modified.Since the designof the finite-time velocity observer is finished, the globalfinite-time is changed as Ttol′ = Ttol + Tc=Ta + Tb + Tc.

In order to guarantee that the system achieves globalstability in finite time Ttol′, the modified input torque τ isproposed as follows:

τ =

{cτ 0 ≤ t < Tc

τ t ≤ Tc(16)

where cτ ∈ R3×1 is an arbitrary continuous function.Hence, based on the saturation function in (7), the control

input u is given by

u =

uupper if τ > uupper

τ if −ulower ≤ τ ≤ uupper−ulower if τ < −ulower

(17)

Therefore, the closed-loop control system of the 3-DOFmanipulator, which is composed of the disturbance observer,velocity observer, and the nonlinear controller, is establishedto guarantee the trajectory tracking in finite time Ttol′.

IV. GLOBAL FINITE-TIME STABILITYBy using Lyapunov theory, this section proves the globalfinite-time stability of the closed-loop control system. That isto say, the finite-time tracking toward the desired trajectorycan be testified.

Two lemmas are introduced to contribute to the demonstra-tion of global finite-time stability.

Lemma 1 [11]: Consider a nonlinear system x (t) = f (t),x ∈ Rn, x (0) = x0 with the equilibrium point x = 0.Assume that there exist three real numbers ρ1 > 0, ρ2 > 0,0 < β < 1 and a continuously differentiable positivefunction V (x) : Rn → R such that the inequality V (x) +ρ1V (x) + ρ2V

β (x) ≤ 0 is always fulfilled for any solutionx (t, x0) of the system. Then, the equilibrium point x = 0 isglobally finite-time stable and the finite convergence time T ,which is called the finite settling time, satisfies the inequalityT (x0) ≤ (ρ1 (1− β))

−1 (ln(ρ1V

1−β (x0) + ρ2)− ln ρ2

).

Lemma 2 [29]: Consider a double integrator system isexpressed by differential equations x = v, v = u, whereu is the control input, and x and v are the state variables.This system can be made globally finite–time stable byimplementing control input (18).

u=−|v|$sgn(v)−|ψ (x, v) |$(2−$)−1

sgn (ψ (x, v))

ψ (x, v) = x+ (2−$)−1 |v|2−$sgn (v) .

(18)

where tuning coefficient 0 < $ < 1 can be used to adjust thefinite convergence time. Furthermore, by using control input(18), state variables x and v exactly converge to zero in thefinite convergence time specified as

T (x (0) , v (0)) ≤ V (x (0) , v (0))F

Fl,

F = (3−$)−1

(1−$)(19)

where positive function V (x, v) and coefficient l are definedsuch that arbitrary parameters ζ and Φ fulfill constraints 0 <ζ < 1 and Φ > 1, respectively [29].

V =1

J|ψ (x, v) |J + ζvψ (x, v) +

Φ

3−$|v|3−$

l = −max(x,v)∈D V (x, v) , D = {(x, v) : V (x, v) = 1}(20)

where J = (2−$)−1

(3−$).Theorem 1 demonstrates that the tracking error dynamicsexpressed by (8) reach sliding surface s = 0 in finite timeTa. Theorem 2 shows that the estimation of disturbances τdconverges to actual disturbances s = 0 in finite time Ta. The-orem 3 proves that the tracking errors, which had reached s =0, arrive at zero in finite time Tb. Moreover, Theorem 4 showsthat the estimations of velocity converge to actual velocityin finite time Tc. Finally, Theorem 5 shows that trajectorytracking is achieved within finite time T

′

tol = Ta + Tb + Tc.Theorem 1: Consider control torque (12) and sliding mode

(11). Then, the tracking errors expressed by (8) reach slidingsurface s = 0 within adjustable finite time Ta given by (10).

Proof: Consider candidate Lyapunov function V =0.5(sT s+ ωTω

). By substituting for τ from (12) into (8),

one can obtain the following equation:

ev = φ− sig$x (ev)− sig$v (ψ (ex, ev)) + τs (21)

Then, by substituting ev from (21) into (11), sliding surfaces is derived as s = φ + τs + ω so that s = φ + τs + ω. Bytaking into account ω and τs expressed by (9) and (12), thisyields

s = φ− φ− εs− ςsigc (s)−(β0 (t)λmax

(M−1

)+β2 (t)

)sgn (s)

(22)

By employing inequalities ‖s‖≤3∑j=1

|sj |, sT φ≤‖s‖‖φ‖, and

−sTφ ≤‖s‖‖φ‖, it follows that

sT s≤‖s‖‖ φ‖+‖s‖‖φ‖ −3∑j=1

ςj |sj |c+1

−3∑j=1

εjs2j−(β0(t)λmax

(M−1

)+β2(t)

)‖s‖

(23)

According to the definition φ = −p (x) τd, inequalities (2),and inequality M−1 ≤ λmax

(M−1

)[14], the following

inequalities are obtained:

‖ φ ‖≤ β2 (t) , ‖ φ ‖≤ β0 (t)λmax

(M−1

)(24)

where function β2 (t) is defined as β2 (t) = β0 (t) ‖ ˙p (x) ‖+β1 (t)λmax

(M−1

).

Applying (24) to (23) results in

sT s ≤ −εmin

3∑j=1

s2j − ςmin

3∑j=1

|sj |c+1 (25)

where εmin = min εj and ςmin = min ςj . Based on (9), thederivation of ω is obtained as follows:ω = z − ev = −Kω + pu− vd − ςsigc (ω)− ev−β0(t)λmax

(M−1

)sgn(ω)−|f(x, v)|sgn(ω)

(26)

4



Applying (8) to (26), this yields

ω = −Kω − β0 (t)λmax

(M−1

)sgn (ω)

−γsigc (ω)−|f (x, v) |sgn(ω)−f (x, v)−φ. (27)

Thus,

ωT ω= −ωT (f (x, v) + φ+ γsigc (ω) +Kω)

−β0(t)λmax

(M−1

)( 3∑j=1

|ωj |

)−|ω|T |f(x, v)|. (28)

Consider that −|ω|T |f (x, v) | −ωT f (x, v) < 0 , −ωTφ ≤

‖ ω ‖ ‖ φ ‖ and ‖ ω ‖≤3∑j=1

|ωj | are always fulfilled. Thus,

ωT ω satisfies the following inequality:

ωT ω ≤ −3∑j=1

kjω2j −

3∑j=1

γj |ωj |c+1

+‖ω‖(‖φ‖−β0(t)λmax

(M−1

)) (29)

By employing ‖φ‖≤β0(t)λmax

(M−1

), (29) can be simpli-

fied to

ωT ω ≤ −kmin

3∑j=1

ω2j − γmin

3∑j=1

|ωj |c+1 (30)

Taking into account (30) and (25), and applying definitionsα = min (min εj ,min kj) and δ = min (min γj ,min ςj),derivation V can be written as

V = 0.5(sT s+ ωT ω

)≤ −α

3∑j=1

(ω2j +s2j

)−δ

3∑j=1

(|ωj |c+1+|sj |c+1

) (31)

Introducing inequalities3∑i=1

|ai|β ≥(

3∑i=1

|ai|)β

and

3∑i=1

|ai|1+β ≥

√(3∑i=1

|ai|2)1+β

for 0 < β < 1 results in

V ≤ −2αV − δ

√√√√( 3∑

j=1

(s2j + ω2

j

))c+1

= −2αV − δ√

2c+1√V c+1

(32)

Suppose ρ1 = 2α, ρ2 = δ√

2c+1, and β = 0.5 (c+ 1). Thus,V (x) + ρ1V (x) + ρ2V

β (x) ≤ 0. According to Lemma 1,the equilibrium point s = 0 is stable and ω = 0 is attainableafter finite time Ta. This completes the proof.

Theorem 2: Consider disturbance observer (9). Then, dis-turbance errors converge to zero within finite time Ta givenby (10).

Proof: From Theorem 1, it is known that ω=0 is attainablefor t ≥ Ta such that ω = 0. By setting φ = φ−φ, this yields

φ=−φ−Kω−f(x, v)−β0(t)λmax

(M−1

)sgn(ω)

−γsigc (ω)−|f (x, v) |sgn (ω)=−φ−Kω+φ−ev+p (x)u−vd−β0 (t)λmax

(M−1

)sgn(ω)

−γsigc (ω)−|f(x, v)|sgn (ω)= z−ev= ω

(33)

Clearly, for t ≥ Ta, φ = 0, as a result, φ = φ. Thus, afterfinite time Ta, τd = p−1φ converges to τd = p−1φ. Thiscompletes the proof.

Theorem 3: Consider tracking error dynamics (8) andsliding mode (11). Then, trajectory tracking errors, whichhad arrived at s = 0, converge to zero within finite time Tbspecified as

Tb ≤ maxj{ 1

ljFj(Vj (exj (Ta) , evj (Ta)))

Fj}

with Fj = (3−$xj)−1

(1−$xj)(34)

where positive function Vj (exj , evj) and lj are defined sothat parameters ζj and Φj follow 0 < ζj < 1 and Φj > 1,respectively.

Vj=|ψj (exj , evj) |Jj

Jj+ζjevjψj (exj , evj)+

Φj |evj |3−$xj3−$xj

,

lj=−max(exj ,evj)∈Dj Vj (exj , evj),Dj={(exj , evj) :Vj (exj , evj)=1} .

(35)

where Jj = (2−$xj)−1

(3−$xj).Proof: According to Theorem 1, s = ω = 0 for t ≥ Ta.

So, from (11) one can obtain

ev=−sig$x (ev)−sig$v (ψ (ex, ev)) . (36)

Therefore, (8) can be rewritten as

ex = evev = −sig$x (ev)− sig$v (ψ (ex, ev))

(37)

In consequence, three independent double integrator subsys-tems that belong to system (37) are described as below forj = 1, 2, 3:

exj=evjevj =−|evj |$xjsgn (evj)−|ψj(exj , evj) |$vjsgn(ψj(exj , evj))

(38)Based on Lemma 2 and the conclusion of [29], the followinginequality can be written as

Vj (exj , evj) ≤ lj (Vj (exj , evj))

2

3−$xj ,

for all [exj , evj ]T ∈ R2.

(39)

where lj = −max(exj ,evj)∈Dj Vj (exj , evj). As a result,tracking errors in each subsystem converge to zero after finitetime jTb specified as

jTb ≤1

ljFj(Vj (exj (Ta) , evj (Ta)))

Fj (40)

Obviously, for system (37) all tracking errors that had arrivedat s = 0 converge to zero after finite time Tb. In otherwords, tracking errors converge to zero within finite timeTtol = Ta + Tb.

Proposition 1: f (x, v) ∈ R3, which is defined in (6),satisfies the following relation:

‖f (x, v)− f (x, v)‖≤ L1 (x, v) ‖ v − v‖ +L2 (x, v) (41)

5



Proof: From (6), this yields

‖f (x, v)−f (x, v)‖=‖M−1(B (〈v, v〉−〈v, v〉)+C

(v2−v2

))‖

≤‖M−1B (〈v, v〉 − 〈v, v〉) ‖ + ‖M−1C(v2 − v2

)‖

≤ λmax

(M−1

)(‖B ‖(‖〈v, v〉‖+‖〈v, v〉‖)+‖C ‖

(‖v2 ‖+‖ v2 ‖

))(42)

Based on the definitions of 〈v, v〉 and v2, the followinginequalities hold:

‖〈v, v〉‖≤‖v‖2, ‖v2 ‖≤‖v‖2 (43)

So, (42) can be rewritten as

‖f (x, v)−f (x, v)‖

≤ λmax

(M−1

)(‖B ‖

(‖v‖2 +‖ v‖2

)+‖C ‖

(‖v‖2 +‖ v‖2

))≤ λmax

(M−1

) (2 (‖B‖+‖C‖) ‖ v‖2

)+λmax

(M−1

)(‖B‖+‖C‖) (‖v‖−‖ v‖) (‖v‖+‖ v‖)

(44)Then, in view of (13), one can obtain

‖f (x, v)−f (x, v)‖

≤ λmax

(M−1

)(2 (‖B‖+‖C‖) ‖ v‖2 +(‖B‖+‖C‖) (‖ v−v‖) (‖v‖+κ)

)= L1(x, v)‖ v − v‖+L2 (x, v) .

(45)This completes the proof.

Theorem 4: Consider velocity observer (14) and assumethat estimation errors should not have a finite escape time.Then, estimates x and v converge to x and v exactly withinfinite time Tc given by (15).

Proof: Based on (14) and the non-existence of a finiteescape time in estimation errors, the error dynamics can bewritten as follows:

˙ex= ev−(σ0+σ3) sig(ex)−h(x, v) sgn(ex)−σ2sigξ (ex)

˙ev=−σ1v−σ3|ex|+f(x, v)−f(x, v)−φ (x, v, t)(46)

Here, the candidate Lyapunov function is chosen as V =‖ ex ‖ + ‖ ev ‖, such that its time derivative V =‖ ex ‖−1eTx ˙ex+ ‖ ev ‖−1 eTv ˙ev where (ex)j is the jth element of

ex. Introducing inequalities3∑i=1

|ai|β ≥(

3∑i=1

|ai|)β

and

3∑i=1

|ai|1+β ≥

√(3∑i=1

|ai|2)1+β

for 0 < β < 1 results in

3∑j=1

| (ex)j | ≥‖ ex ‖,3∑j=1

| (ex)j |ξ+1 ≥‖ ex ‖ξ+1 (47)

By substituting for ˙ex and ˙ev from (46) into V =‖ ex ‖−1eTx ˙ex+ ‖ ev ‖−1 eTv ˙ev , and employing (47), one can obtain

V ≤‖ ex ‖−1eTx ev−(σ0+σ3)‖ex‖−h (x, v)−σ1 ‖ ev ‖

−σ2 ‖ex‖ξ−σ1 ‖ ev‖−1 eTv v−σ3‖ ev ‖−1eTv |ex|

−‖ ev‖−1eTv φ+‖ ev ‖−1eTv (f(x, v)−f(x, v))

(48)

Applying inequalities | eTv (f (x, v)− f (x, v)) | ≤ ‖ eTv ‖‖f(x, v) − f(x, v)‖, |eTx ev|≤‖ex‖‖ev‖, |eTx v|≤‖ ex ‖‖ v ‖,|eTv |ex|| ≤‖ ev ‖‖|ex|‖ and |eTv φ| ≤‖ ev ‖‖φ‖ results in

V≤‖ev‖−(σ0+σ3)‖ ex ‖−h(x, v)−σ1‖ev‖+‖φ‖

+σ1‖v‖−σ2‖ex‖ξ+σ3‖ex‖+‖f(x, v)−f(x, v)‖(49)

By employing inequalities (13) and (24), equality σ1 = 1 +σ0 and proposition 1, and substituting for h (x, v), (48) canbe rewritten as

V ≤ −σ0 ‖ ev ‖ −σ0 ‖ ex ‖ −σ2 ‖ ex ‖ξ

−σ2 (‖ v‖+κ)ξ−L2 (‖ v‖+κ) + L2 ‖ ev ‖

(50)

Then, by incorporating with inequalities ‖ ev ‖≤ (‖ v‖ +κ)and − (‖ v‖ +κ)

ξ ≤ − ‖ ev ‖ξ, this yields

V ≤−σ0(‖ ex ‖+‖ ev ‖)− σ2(‖ ex ‖ξ+‖ ev ‖ξ

)(51)

According to inequality3∑i=1

|ai|β ≥(

3∑i=1

|ai|)β

, ‖ ex ‖ξ

+ ‖ ev ‖ξ≥ (‖ ex ‖ + ‖ ev ‖)ξ is obtained.

Moreover, in view of function V =‖ ex ‖ + ‖ ev ‖, (51) canbe rewritten as V + σ0V + σ2V

ξ ≤ 0. By setting ρ1 = σ0,ρ2 = σ2, β = ξ, inequality V (ex, ev) + ρ1V (ex, ev) +ρ2V

β (ex, ev) ≤ 0 is satisfied. Considering Lemma 1, errorsex and ev reach zero for t ≥ Tc.This completes the proof.

Theorem 5: Consider a closed loop system of a 3-DOFmanipulator, which only measures positions of joints. Then,joint positions x converge to the desired trajectories xd withinfinite time T

′

tol = Ta +Tb +Tc, where Tc satisfies inequality(15), and Tb and Ta are determined by

Tb ≤ maxj{Vj (exj (Ta + Tc) , evj (Ta + Tc))

Fj

ljFj} (52)

Ta≤

ln

α

√√√√( 3∑j=1

s2j (Tc)+ω2j (Tc)

)1−c

+δ

−ln δ

α (1− c)

(53)

Proof: Based on Theorem 4, estimate v converges to v afterfinite time Tc using the velocity observer. Note that positionx is applied to observer (14) as input to estimate v, and fort ≥ Tc improved torque τ becomes τ , and v becomes v.Consequently, by considering Theorem 1, 2, 3 and 4, it isclear that x converges to xd within finite time T

′

tol = Ta +Tb + Tc, where Tc satisfies inequality (15). In addition, it isclear that initial condition t = 0 in (10) could be substituted

6



by t = Tc, and initial condition t = Ta in (33) could bereplaced by t = Ta + Tc. This completes the proof.

Remark 2: The constraint of input torque indicated in (17),which is employed to design control signal for eliminatingthe effect of actuator saturation, can enhance the performanceof the control system of manipulator.

Remark 3: The parameters of nonlinear observers given in(9) and (14) as well as those of control torque provided in(12), which are in proper design, can decrease global finitetime and improve control precision.

V. NUMERICAL SIMULATIONIn this section, finite-time trajectory tracking is verified bynumerical simulation for the 3-DOF manipulator describedin Section 2.

A schematic block diagram of the control system of theclosed-loop 3-DOF manipulator is given as Fig. 2.

FIGURE 2. Closed-loop control system of 3-DOF manipulator.

Consider the model of the 3-DOF manipulator in (1) andthe matrix parameters in (3). The desired trajectory was cho-sen as xd = [sin (8t) , cos (8t) , 0.5 cos (8t)]

T . The initial po-

sition and velocity were selected as x (0) =[−π

3,−π, π

3

]Tand v (0) = [0, 0, 0]

T , respectively. The initial values of thevelocity observer were considered to be x (0) = [0, 0, 0]

T

and v (0) = [5,−6,−3]T . The disturbance matrix was

chosen as τd = (1 + sin (t)) [1, 1, 1]T

+ 0.2x (t). Basedon (17), values of uupper1 = uupper2 = uupper3 = 80and ulower1 = ulower2 = ulower3 = 60 were selected.The upper bound of velocity was chosen as κ = 25. Theparameters of the sliding mode controller were taken to be$x1 = $x2 = $x3 = 0.5, ε1 = ε2 = ε3 = 5 andς1 = ς2 = ς3 = 2. The parameters of the velocity observerwere assumed to be σ0 = σ2 = σ3 = 1.5, and ξ = 0.8,and the disturbance parameters were chosen as c = 0.2,γ1 = γ2 = γ3 = 15, and k1 = k2 = k3 = 10.

From Fig. 3, it is clear that the estimated velocities v1 (t),v2 (t), and v3 (t) converge to actual velocities v1 (t), v2 (t),and v3 (t) in finite time Tc ≈ 4.2 (s). In Fig. 4, the estimateddisturbances τd1, τd2, and τd3 converge to actual disturbancesτd1, τd2, and τd3 within finite time Ta ≈ 1.0 (s). Fig. 5illustrates how the actual positions x1, x2, and x3 trackexactly the desired positions xd1, xd2, and xd3 in total finitetime T

′

tol ≈ 8.5 (s).As shown in Fig. 3, 4, and 5, global finite-time stability in

a closed-loop system of a 3-DOF manipulator is attainable,consistent with the results of the theoretical analysis.

(a)

(b)

(c)

FIGURE 3. The estimates of joints velocities. a: Time responses of v1 and v1.b: Time responses of v2 and v2. c: Time responses of v3 and v3.

(a)

(b)

(c)

FIGURE 4. The estimates of external disturbances. a: Time responses of τd1and τd1. b: Time responses of τd2 and τd2.c: Time responses of τd3 and τd3.

7



(a)

(b)

(c)

FIGURE 5. Finite time trajectory tracking. a: Time responses of xd1 and x1.b: Time responses of xd2 and x2. c: Time responses of xd3 and x3.

VI. CONCLUSIONThis paper addresses the finite-time trajectory tracking con-trol problem for the rigid 3-DOF manipulator system inthe presence of system external disturbance. The finite-timevelocity observer is designed to estimate velocity from directmeasurement of joint position. To improve the robustnessof the manipulator system against the disturbances, a distur-bance observer is also designed. By applying the disturbanceobserver and velocity observer with a nonlinear controller,the conditions of global finite-time stability is established.Based on Lyapunov stability theory, it is proved that thefinite-time trajectory tracking is achieved for the closed-loop manipulator system. Simulation is performed and theresults demonstrate the effectiveness of the proposed controlmethod. In the future, the finite-time trajectory tracking con-trol based on the observers will be further investigated forother nonlinear systems. In addition, the finite-time conver-gence rate for the robot system will be studied by virtue ofnew control schemes.

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ZIYANG CHEN received the B.E. degree in Elec-trical engineering and its automation from South-west Petroleum University , Chengdu, China, in2017. Now, he is pursuing the M.E. degree inControl engineering in Nan Chang University. Heresearch interests include robot and control sys-tem.

XIAOHUI YANG received the B.Sc. M.Sc., andPh.D. degrees from Nanchang University, Nan-chang, China, in 2003, 2006, and 2015, respec-tively (email: [email protected]). He hasbeen with the Department of Electronic Informa-tion Engineering, School of Information Engineer-ing, Nanchang University, since 2006, where he iscurrently an Associate Professor. He has publishedover 30 research articles. His current interestsinclude intelligent control, process control, fault

diagnosis and stochastic nonlinear systems etc.

XIAOLONG ZHANG received the B.E. degreein the Electrical engineering and its automationfrom Hefei Normal University, Hefei, China, in2017, Now, he is pursuing the M.E. degree in elec-tric engineering in Nanchang University, His re-search interests include dynamics and simulationof mechanical systems, robotics and constrainedsystems .

PETER X. LIU is with the Department ofSystems and Computer Engineering, CarletonUniversity, Ottawa, K1S 5B6 Canada (email:[email protected]). He received his B.Sc. andM.Sc. degrees from Northern Jiaotong University,China in 1992 and 1995, respectively, and Ph.D.degree from the University of Alberta, Canada in2002. He has been with the Department of Sys-tems and Computer Engineering, Carleton Univer-sity, Canada since July 2002 and he is currently

a Professor and Canada Research Chair. He is also with the School ofInformation Engineering, Nanchang University as an Adjunct Professor. Hisinterest includes interactive networked systems and teleoperation, haptics,micro-manipulation, robotics, intelligent systems, context-aware intelligentnetworks, and their applications to biomedical engineering. Dr. Liu has pub-lished more than 280 research articles. He serves as an Associate Editor forseveral journals including IEEE Transactions on Cybernetics, IEEE/ASMETransactions on Mechatronics, IEEE Transactions on Automation Scienceand Engineering, and IEEE Access. He received a 2007 Carleton ResearchAchievement Award, a 2006 Province of Ontario Early Researcher Award,a 2006 Carty Research Fellowship, the Best Conference Paper Award of the2006 IEEE International Conference on Mechatronics and Automation, anda 2003 Province of Ontario Distinguished Researcher Award. Dr. Liu is alicensed member of the Professional Engineers of Ontario (P.Eng), a seniormember of IEEE and Fellow of Engineering Institute of Canada (FEIC).

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Finite-time trajectory tracking control for rigid 3-DOF ... · results of a 3-DOF manipulator shows that trajectory tracking was achieved with a ﬁnite period of settling time. The