J Intell Robot SystDOI 10.1007/s10846-013-9888-5

Adaptive Neural Network Finite-Time Controlfor Uncertain Robotic Manipulators

Haitao Liu · Tie Zhang

Received: 5 April 2013 / Accepted: 10 September 2013© Springer Science+Business Media Dordrecht 2013

Abstract An adaptive neural network finite-timecontroller (NNFTC) for a class of uncertain non-linear systems is proposed by using the backstep-ping method, which employs an adaptive neuralnetwork (NN) system to approximate the struc-ture uncertainties and uses a variable structureterm to compensate the approximation errors,thus improving the robustness of the system toexternal disturbances. The controller is then ap-plied to uncertain robotic manipulators, with acontrol objective of driving the system state to theoriginal equilibrium point. It is proved that theclosed-loop system is finite-time stable. Moreover,simulated and experimental results indicate thatthe proposed NNFTC is effective and robust.

Keywords Finite-time control · Neural network ·Robotic manipulator · Uncertainty

H. Liu (B)Guangdong Ocean University, Zhanjiang, 524088Guangdong, Chinae-mail: gdliuht@126.com

T. ZhangSouth China University of Technology, Guangzhou,510640 Guangdong, China

1 Introduction

Asymptotic stability implies that the trajectory ofa system converges to the equilibrium as time goesto infinity, while finite-time stability of dynamicsystems may give rise to fast, transient and high-precision performance besides finite-time conver-gence to the equilibrium. Therefore, continuousfinite-time control attracts much interesting fromresearchers, and a great deal of work has beencarried out [1–9].

Although the finite-time control is very use-ful for high-performance robotic manipulator sys-tems, few researches on its application in roboticmanipulators were carried out. Hong et al. [10]firstly proposed a finite-time controller for robotsystems with state feedback and dynamic out-put feedback control. Su [11] proposed a globalfinite-time tracking controller by replacing thelinear errors in linear proportional-derivative plusscheme with nonsmooth but continuous expo-nential errors. Zhao et al. [12] presented a ro-bust finite-time stability control approach for ro-bot systems by means of backstepping. Su andZheng [13] implemented the global finite-timetracking of robot manipulators by eliminatingnonlinear exponential-like errors. Rubio et al.[14, 15] discussed the optimal control of roboticarms based on a mathematical model. All theseabove-mentioned researches are based on theassumption that the dynamic model of robotic

J Intell Robot Syst

manipulator is exactly known. However, in prac-tice, the precise dynamic model of robotic ma-nipulator is impossible to establish due to theinfluences of various uncertain factors [16], so thatthe robustness of the finite-time control systemcan not be guaranteed. In order to solve thisproblem, Yu et al. [17] proposed a robust contin-uous finite-time controller for robotic manipula-tors with terminal sliding mode. Rubio et al. [18]put forward a sliding model controller for roboticarms without knowing the dynamics of the arms,but the closed-loop system is only asymptoticallystable.

In recent years, neural networks have beenpaid more and more attention to in the con-trol of nonlinear systems owing to their non-linear approximation and learning abilities [19],and have gained a lot of successful applications[20–24]. Some researchers dealt with the neuralnetwork-based compensation for robotic systemsto eliminate the effect of uncertainties [25–29].As these above-mentioned neural network-basedcontrollers only provided asymptotical stabilitybut not finite-time stability, Wang et al. [30] de-signed a neural network-based terminal sliding-mode controller of robotic manipulators consid-ering actuator dynamics, by which the finite-timeconvergence and stability of the closed-loop sys-tem can be guaranteed. Liu et al. [31] proposeda novel neural network-based robust finite-timecontrol strategy of robotic manipulators, whichguaranteed excellent finite-time convergence andstrong robustness.

In this paper, we study the finite-time stabilityof robotic manipulators with structured and un-structured uncertainties by using the backsteppingmethod, and propose an adaptive neural networkfinite-time controller (NNFTC) to improve the ro-bustness of the closed-loop system. In the NNFTCthat consists of a finite-time control module, aRBF neural network (NN) approximation moduleand a variable structure control module, NN isto adaptively approximate the structure uncer-tainties, and a variable structure term is used tocompensate the approximation errors. With thehelp of NNFTC, the tracking errors converge tozero in finite time, which resulting in fast response,strong robustness and high precision of the closed-loop system. Certainly, the proposed finite-time

controller is different from the terminal sliding-mode controller presented in [30] and our previ-ous work presented in [31].

The rest of this paper is organized as follows:In Section 2, the preliminaries about the finite-time stability theorems and the related lemmasare introduced. In Section 3, the NNFTC for aclass of uncertain nonlinear systems is proposedby means of backstepping and the finite-time sta-bility is proved. Section 4 discusses the applicationof the NNFTC to robotic manipulators. Section5 presents the simulated and experimental resultsto demonstrate the effectiveness of the proposedcontroller. And, finally, Section 6 draws someconclusions.

2 Preliminaries

The concepts related to finite-time control aregiven in [32]. In addition, we also recall the cor-responding Lyapunov stability theorem for finite-time stability of autonomous systems [4], whichwill be used in the next section.

Definition 2.1 Considering a time-invariant non-linear system in the form of.x = f (x) , f (0) = 0, x ∈ Rn (1)

where f : U0 → Rn is continuous in an openneighborhood U0 of the origin, the equilibriumx = 0 of the system is (locally) finite-time stableif:

(1) it is asymptotically stable in U , an openneighborhood of the origin, with U ⊆ U0;

(2) it is finite-time convergent in U , that is, forany initial condition x0 ∈ U\ {0}, there is asettling time T > 0 such that every solutionx(t, x0) of system (1) is defined with x (t, x0) ∈U\ {0} for t ∈ [0, T) and satisfies

limt→T

x (t, x0) = 0 (2)

And, if t ≥ T, x(t, x0) = 0. Moreover, if U = Rn,the origin x = 0 is globally finite-time stable.

Clearly, global asymptotic stability and localfinite-time stability imply global finite-time sta-bility. In addition, finite-time stability implies theuniqueness in forward time of the solution x = 0.

J Intell Robot Syst

Definition 2.2 Considering a nonlinear system inthe form of

.x = f (x, u) , f (0, 0) = 0, x ∈ Rn, u ∈ Rm

(3)

where f : Rn×m → Rn is globally defined, the ori-gin of system (3) is finite-time stable if there is astate feedback control law in the form of u = φ(x)with φ(0) = 0 such that the origin of the closed-loop system

.x = f (x, ϕ (x)) is in finite-time stable

equilibrium state.

Lemma 2.3 [2] Considering the nonlinear system(1) and supposing that there is C1 function V(x)

def ined in neighborhood U ⊂ Rn of the origin, andthat real numbers c > 0 and 0 < α < 1, such that

(i) V(x) is positive-def inite in U;(ii)

.

V (x) + cVα (x) ≤ 0, ∀x ∈ U .

Thus, the origin of system (1) is locally f inite-time stable. The settling time, which depends on theinitial state x(0) = x0, satisf ies

Tx (x0) ≤ V (x0)1−α

c (1 − α)(4)

for all x0 in some open neighborhoods of the ori-gin. If U = Rn and if V(x) is also radially un-bounded (i.e.V(x) → +∞ as || × || → +∞), theorigin of system (3) is globally f inite-time stable.

Lemma 2.4 [17] For any real number li, i =1, . . . , n, when 0 < γ < 1 and 0 < λ < 2, the fol-lowing inequalities hold:

(|l1| + · · · + |ln|)γ ≤ |l1|γ + · · · + |ln|γ (5)

(|l1|2 + · · · + |ln|2

)λ ≤ (|l1|λ + · · · + |ln|λ)2

(6)

For the convenience of control design and analysis,we def ine vector Sig (·)α ∈ Rn as follows:

Sig (ξ)α = [|ξ1|α sgn (ξ1) , · · · , |ξn|α sgn (ξn)]T (7)

where ξ = [ξ1, · · · , ξn]T ∈ Rn, 0 < α < 1, andsgn(·) is the sign function, which is def ined asfollows:

sgn (ξ) =⎧⎨⎩

1, ξ > 00, ξ = 0−1, ξ < 0

(8)

3 Adaptive Neural Network Finite-Time Control

3.1 Finite-Time Stability

In this paper, we consider a class of second-orderuncertain nonlinear dynamical system{ .

x 1 = x2.x 2 = f (x) + w (x) + g (x) u

(9)

where x = [x1, x2]T is the system state vector, x1 ∈Rn, x2 ∈ Rn and x ∈ R2n. f (x) ∈ Rn and g(x) ∈Rn × n are known smooth nonlinear functions of x,and g(x) is nonsingular on R2n. w(x) ∈ Rn repre-sents the uncertainties and external disturbancessatisfying ||w(x)|| ≤ d, where d > 0, and u ∈ Rn

is the control input. In [33, 34], f (x) and g(x)

are considered as unknown functions, but in thisstudy, w(x) is unknown. Based on the backstep-ping method, a robust finite-time controller is de-signed for the nonlinear dynamic system (9) withuncertainties and external disturbances. More-over, the proof is constructive and it gives a sys-tematic procedure for the design of global finite-time stabilizers for the nonlinear system (9).

Introducing an auxiliary controller ϕ(x1) ∈ Rn

with ϕ(0) = 0, and defining an error variable

z = x2 − ϕ (x1) (10)

Substituting it into Eq. 9, the dynamic system (9)is rewritten as{ .

x 1 = z + ϕ (x1).z = f (x) + g (x) u + w (x) − .

ϕ (x1)(11)

In the following, we design the state feedbackcontroller u using a backstepping procedure.

Step 1: Differentiating the Lyapunov function

V1 = 12

xT1 x1 (12)

J Intell Robot Syst

with respect to time, one can obtain

.

V 1 = xT1

.x 1

= xT1 (z + ϕ (x1)) (13)

In order to obtain finite-time stability, Eq. 13should satisfies the conditions of Lemma 2.3,so that we define an auxiliary control ϕ (x1) =−K1Sig (x1)

α , where K1 = diag(k11, k12, . . . , k1n)

and 0 < α < 1. Substituting the auxiliary controlinto Eq. 13, there comes

.

V 1 = −xT1 K1Sig (x1)

α + xT1 z (14)

If z = 0, one can obtain

.

V 1 = −xT1 K1Sig (x1)

α

= −n∑

i=1

k1i |x1i|1+α

≤ −k1

(12

n∑i=1

x21i

)μ

≤ −k1Vμ

1 (15)

where μ = (1 + α)/2, 1/2 < μ < 1, k1min = min{k1i} and k1 = 2μk1min.

Therefore, the next design step is needed.Step 2: For the Lyapunov function

V2 = V1 + 12

zTz (16)

its derivative is

.

V 2 = .

V 1 + zT .z

= − xT1 K1Sig (x1)

α +xT1 z+zT

× (f (x) + g (x) u + w (x) − .

ϕ (x1))

= − xT1 K1Sig (x1)

α +zT

× (x1 + f (x) + g (x) u + w (x) − .

ϕ (x1))(17)

In order to satisfy the conditions of Lemma 2.3(ii); consequently, the controller u is designed as

u = g−1 (x)(

.ϕ (x1) − f (x) − w (x) − x1

−K2Sigα (z))

(18)

where K2 = diag(k21, k22, . . . , k2n). The substitu-tion of Eq. 18 into Eq. 17 yields

.

V 2 = .

V 1 + zT .z

= −xT1 K1Sig (x1)

α − zT K2Sig (z)α

≤ −k1Vμ

1 − k2

(12

n∑i=1

z2i

)μ

≤ −kVμ

2 (19)

where k2min = min{k2i}, k2 = 2μk2min and k =min

{k1, k2

}.

Clearly, according to Lemma 2.3, the closed-loop system of Eqs. 11 and 18 is finite-time stable.But the uncertain function w(x) is unknown, sothat the controller (18) can not be applied inpractice. Therefore, a variable structure term isintroduced to improve the robustness to uncer-tainties w(x).

Theorem 3.1 For the uncertain nonlinear dynami-cal system (9), if we design the following controller:

u = g−1 (x)(

.ϕ (x1) − f (x) − x1 − K2Sigα (z)

−K3sgn (z))

(20)

where K3 = diag(k31, k32, . . . ,k3n), the closed-loopsystem is globally f inite-time stable.

Proof Substituting Eq. 20 into Eq. 17, there is

.

V 2 = .

V 1 + zT .z

= −xT1 K1Sig (x1)

α − zT K1Sig (z)α

+zT (w − K3sgn (z)) (21)

J Intell Robot Syst

If k3i ≥ d, there comes

.

V 2 = .

V 1 + zT .z

= −xT1 K1Sig (x1)

α − zT K1Sig (z)α

+zT (w − K3sgn (z))

= −xT1 K1Sig (x1)

α − zT K1Sig (z)α

−zT (K3sgn (z) − w)

≤ −xT1 K1Sig (x1)

α − zT K1Sig (z)α

≤ −k1Vμ

1 − k2

(12

n∑i=1

z2i

)μ

≤ −kVμ

2 (22)

According to Lemma 2.3, the closed-loop systemis global finite-time stable, and the setting timeestimation is

T (x) ≤ 1

k (1 − μ)V2 (x0)

1−μ (23)

Thus, the nonlinear system (9) with control input(18) possesses faster response, stronger robustnessand higher precision than asymptotically stablesystems. �

Remark 1 It is obvious that the derivative ofSig(x1)

α is infinite when x1i = 0 and.x 1i �= 0 in

.ϕ (x1). To avoid this singularity problem, it is nec-essary to set the threshold value λ to judge singu-larity, so that the following definition of

.ϕ (x1) is

modified:

.ϕ i (x1i)=

⎧⎨⎩−k1iα|x1i|α−1 .

x 1i,if |x1i|≥λ and.x 1i �= 0

−k1iα|�i|α−1 .x 1i,if |x1i|<λ and

.x 1i �= 0

0, if.x 1i = 0

(24)

where λ and �i are both small positive constants,x1i is the is the ith element of vector x1, and

.ϕ i (x1i)

is the ith element of vector.ϕ (x1).

Remark 2 As smaller α can make the error con-verges faster and create a “chattering” phenom-enon that may excite unmodeled high-frequency

dynamics, therefore, α should be generally se-lected as 0.7 < α < 0.9.

Remark 3 The variable structure term sgn(z) inEq. 20 is designed to improve the robustness,but it will create “chattering” phenomenon, whichmay excite unmodeled high-frequency dynamicsand even damage the physical device. In orderto eliminate the “chattering”, the saturation func-tion method, which only degrades the robustnessbut does not affect the finite-time stability of theclosed-loop system, is employed. Thus, the signfunction is replaced by the following function:

sat (z) =⎧⎨⎩

sgn(z

δ

), if

∣∣∣zδ

∣∣∣ ≥ 1zδ, others

(25)

where δ is a smaller positive constant calledboundary layer thickness.

The proof is given in Appendix A.

Remark 4 In practice, it is quite difficult to es-timate the supremum of system uncertainties.Therefore, the neural network is adopted to ap-proximate the uncertainties w(x).

3.2 Adaptive Neural Network

A three-layer neural network is employed toadaptively approximate the uncertain term w(x)

and compensate the nonlinear system. The esti-mation w of w can be written as

w = θTφ (x) (26)

where x = [x1, x2]T ∈ R2n is the network inputvector, θ is the estimation of network weight ma-trix θ , n > 1 is the number of the neurons, andφ (x) = [φ1 (x) , φ2 (x) , · · · , φm (x)]T, with φi (·) be-ing the Gaussian RBF functions:

φi (x) = exp

[−‖x − ci‖2

2σ 2i

], i = 1, 2, · · · , n (27)

where ci ∈ Rl and σ i > 0 are center and width ofthe ith basis function, respectively.

J Intell Robot Syst

According to the universal approximation prin-ciple of RBF neural network, the following as-sumptions are given.

Assumption 1 For any given small positive num-ber εN , the optimal weight matrix θ∗ can always befound to make the approximation error

‖ε (x)‖ =∥∥∥θ∗Tφ (x) − w

∥∥∥ < εN (28)

Assumption 2 The optimal weight matrix θ∗ isbounded, i.e. there exist a positive constant λ

satisfying ‖θ∗‖F ≤ λ.Therefore, the nonlinear uncertain function w

can be expressed as

w = θ∗Tφ (x) + ε (29)

Theorem 3.2 For the uncertain nonlinear dynami-cal system (9), if the controller is designed as

u = u1 + u2

u1 = g−1 (x)(

.ϕ (x1) − f (x) − x1 − K2Sigα (z)

−θTφ (x))

u2 ={

−g−1 (x)z

‖z‖ K3, if ‖z‖ �= 0

0, if ‖z‖ = 0(30)

the RBF neural network adaptive law is

.

θ = φ (x) zT (31)

where K3 > 0, is a positive-def inite diagonalmatrix. The closed-loop system satisfying Assump-tions 1 and 2 is robust, adaptive and globally f inite-time stable.

Substituting the control law Eq. 30 into Eq. 9,we can obtain the closed-loop system as follows:

{ .x 1 = −K1Sig (x1)

α + z

z = −K2Sig (z)α − x1 + w − θTφ (x) − K3z

‖z‖(32)

In the following, by using the Backsteppingmethod, the adaptive law is designed, and thefinite-time stability of robot system is also proved.

Step 1: According to the Lyapunov stabilitytheory, the adaptive law (31) is designed, and theasymptotic stability is proved. Therefore, defining

θ = θ∗ − θ , there exists.

θ = .

θ∗ −.

θ = −.

θ .Differentiating the candidate Lyapunov

function

V = 12

xT1 x1 + 1

2zTz + 1

2tr

(θT −1θ

)(33)

along the system (32), there comes

.

V = xT1

.x 1 + zT .

z +tr(θT −1

.

θ)

= xT1 (−K1Sig (x1)

α + z) + zT

×(−K2Sig (z)α − x1+w − θTφ (x)−K3

z‖z‖

)

+ tr(θT −1

.

θ)

= − xT1 K1Sig (x1)

α + xT1 z − zT K2Sig (z)α

− zTx1 + zT(

w − θTφ (x) − K3z

‖z‖)

+ tr(θT −1

.

θ)

= − xT1 K1Sig (x1)

α − zT K2Sig (z)α + zT

×(

θ∗Tφ (x) + ε − θTφ (x) − K3z

‖z‖)

+ tr(θT −1

.

θ)

= − xT1 K1Sig (x1)

α − zT K2Sig (z)α + zT

×(

ε − K3z

‖z‖)

+ zT(θ∗T − θT)

φ (x)

+ tr(θT −1

.

θ)

= − xT1 K1Sig (x1)

α − zT K2Sig (z)α + zTε

− K3 ‖z‖ + zTθTφ (x) + tr(θT −1

.

θ)

= − xT1 K1Sig (x1)

α − zT K2Sig (z)α + zTε

− K3 ‖z‖ + tr(θTφ (x) zT + θT −1

.

θ)

J Intell Robot Syst

≤ − xT1 K1Sig (x1)

α − zT K2Sig (z)α + ‖z‖ εN

− K3 ‖z‖ + tr(θTφ (x) zT + θT −1

.

θ)

= − xT1 K1Sig (x1)

α − zT K2Sig (z)α

− ‖z‖ (K3 − εN) + tr(θT(

φ (x) zT + −1.

θ))

(34)

Therefore, the adaptive law is designed as Eq. 31.Selecting K3 > εN > |ε|, one can obtain

.

V =−xT1 K1Sig (x1)

α − zT K2Sig (z)α

− ‖z‖ (K3 − εN) + tr(θT(

φ (x) zT + −1.

θ))

= − xT1 K1Sig (x1)

α − zT K2Sig (z)α

− ‖z‖ (K3 − εN)

≤ − xT1 K1Sig (x1)

α − zT K2Sig (z)α

≤ 0 (35)

From Eq. 35, we can see that the L2 norms of x1

and z are bounded. According to the definitionof ϕ(x1) and z, x2 is also bounded. For mostactual dynamic systems, assuming that

.x 2 is also

bounded, such as velocity error vector in roboticmanipulators, therefore, according to BarbalatLemma, when t → ∞, x1 → 0,z → 0 and x2 →0,the closed-loop system (32) is asymptoticallystable.

Step 2: Finite-time stabilityFrom Eq. 27, as the Gaussian function φ(x)

is bounded, namely 0 < φ(x) ≤ 1, ||φ(x)|| is alsobounded. Without a loss of generality, setting‖φ (x)‖ ≤ √

m, according to the characteristics

of Frobenius norm,∥∥∥θTφ (x)

∥∥∥F

is also bounded,

that is,

∥∥∥θTφ (x)

∥∥∥F

≤∥∥∥θ

∥∥∥F

‖φ (x)‖ (36)

Differentiating the following Lyapunov function

V = 12

xT1 x1 + 1

2zTz (37)

along the system (32), if K3 > εN +∥∥∥θTφ (x)

∥∥∥F

,

we have.

V = xT1

.x 1 + zT .

z

= xT1 (−K1Sig (x1)

α + z) + zT

×(−K2Sig (z)α − x1+w−θTφ (x)−K3

z‖z‖

)

= −xT1 K1Sig (x1)

α + xT1 z − zT K2Sig (z)α

−zTx1 + zT(

w − θTφ (x) − K3z

‖z‖)

= −xT1 K1Sig (x1)

α − zT K2Sig (z)α

+zT(

θ∗Tφ (x) + ε − θTφ (x) − K3z

‖z‖)

= −xT1 K1Sig (x1)

α − zT K2Sig (z)α

+zT(

ε + θTφ (x) − K3z

‖z‖)

= −xT1 K1Sig (x1)

α − zT K2Sig (z)α

+zT(ε + θTφ (x)

)− K3 ‖z‖

≤ −xT1 K1Sig (x1)

α − zT K2Sig (z)α

+ ‖z‖(εN +

∥∥∥θ

∥∥∥F

‖φ (x)‖)

− K3 ‖z‖= −xT

1 K1Sig (x1)α − zT K2Sig (z)α

− ‖z‖(

K3 − εN −∥∥∥θ

∥∥∥F

‖φ (x)‖)

≤ −xT1 K1Sig (x1)

α − zT K2Sig (z)α

≤ −k1Vμ

1 − k2

(12

n∑i=1

z2i

)μ

≤ −kVμ

2 (38)

where μ=(1+α)/2, 1/2<μ < 1, k1min =min{k1i},k2min = min{k2i}, k1 = 2μk1 min, k2 = 2μk2 min andk = min{k1, k2}.

Therefore, according to Lemma 2.3, for a giveninitial state x(0) = x0, x1 and z will converge to 0in finite time T, T is the settling time. From thedefinition of ϕ(x1) and z, when x1 = 0 and z = 0,x2 = 0, the closed-loop system (32) is also finite-time stable.

Remark 5 The weight of the neural network isadaptive in the Lyapunov stability theorem sense,

J Intell Robot Syst

without requiring convergence to the optimalvalue. Moreover, the variable structure u2 is usedto compensate the neural network’s approxima-tion error, thus guarantee the robustness of thesystem.

Remark 6 When there exists large approximationerror, especially there is abrupt disturbance (suchas load changing), the variable structure termmay create “chattering” phenomenon. In order toeliminate the “chattering”, the variable structureterm is modified as:

u2 =

⎧⎪⎨⎪⎩

−g−1 (x)z

‖z‖ K3, if ‖z‖ ≥ δ

−g−1 (x)z

‖z‖ + δK3, if ‖z‖ < δ

(39)

where δ > 0, δ is the boundary layer thickness.From the above-mentioned proof, it is found thatthe modified variable structure term only de-grades the robustness without affecting the finite-time stability,.

Remark 7 The control parameter K3 should sat-

isfy K3 > εN +∥∥∥θTφ (x)

∥∥∥F

, and is obtained by the

trail and error method.

4 Adaptive Finite-Time Control for RoboticManipulators

In this section, an adaptive finite-time controlleris designed for robotic manipulators, that is,

M (q)..q +C

(q,

.q) .

q +G (q) + τd = τ (40)

where q,.q,

..q ∈ Rn denote the position, veloc-

ity and acceleration, respectively, M(q) ∈ Rn×n isa symmetric and positive-definite inertia matrix,

C(

q,.q)

∈ Rn×n denotes the centrifugal and Cori-olis matrix, G(q) ∈ Rn is the gravity term, τ d ∈Rn represents the bounded unknown disturbancesincluding unstructured unmodeled dynamics, andτ ∈ Rn denotes the torque input vector.

In fact, it is quite difficult or impossible toknown the exact dynamic model of robotic manip-

ulator. Thus, M(q), C(

q,.q)

and G(q), can respec-tively be described as

M (q) = M0 (q) + �M (q)

C(

q,.q)

= C0

(q,

.q)

+ �C(

q,.q)

G (q) = G0 (q) + �G (q) (41)

where M0 (q) , C0

(q,

.q)

and G0(q) are all esti-

mated terms, �M (q) , �C(

q,.q)

and �G(q) areall unknown terms. Then, the dynamic equationof the manipulator can be written in the followingform:

M0 (q)..q +C0

(q,

.q) .

q +G0 (q) + ρ + τd = τ (42)

with

ρ = �M (q)..q +�C

(q,

.q) .

q +�G (q) (43)

Here, the tracking errors e(t) and.e (t) are respec-

tively defined as follows:

e = q − qd,.e = .

q − .q d (44)

where qd is a twice differentiable desired trajec-tory, and q is an actual trajectory. For the sim-plification of notations, let x1 = e, x2 = .

x 1 = .e.

Then the robotic manipulator dynamics with un-certainties is expressed as

{ .x 1 = x2.x 2 = M−1

0 (q)[τ − C0

(q,

.q) .

q −G0 (q)]+w − ..

q d

(45)

where w = −M−10 (q) (ρ + τd) is the uncertain

function including model uncertainties and exter-nal disturbances. The following assumptions aremade about the robot dynamics:

(1) ||M(q)|| < a0;

(2) | ‖ρ‖ < b 0 + b 1 ‖q‖ + b 2

∥∥∥ .q∥∥∥

2.

where a0, b 0, b 1 and b 2 are all positive constants.q and

.q are bounded, so that w is also bounded.

Here, we assume that w ≤ d and d > 0.

J Intell Robot Syst

Fig. 1 Block diagram of the overall closed-loop system

Theorem 4.1 For the robotic manipulator (42)with uncertainties, if the controller is designed as

τ = τ0 + τ1 + τ2

τ0 = M0 (q)( ..

q d − x1 − K2Sig (z)α + .ϕ (x1)

)

+C0

(q,

.q) .

q +G0 (q)

τ1 = −M0 (q) θTφ (x)

τ2 ={

−M0 (q)z

‖z‖ K3, if ‖z‖ �= 0

0, if ‖z‖ = 0(46)

the adaptive law is.

θ = φ (x) zT (47)

where K3 > 0 and is a positive-def inite diagonalmatrix. The closed-loop system of robot is robustand adaptive f inite-time stable.

Remark 8 Substituting Eq. 46 into Eq. 45, wecan conclude that the closed-loop system of therobotic manipulator (45) is the same as Eq. 32.Therefore, it can be easily proved that the closed-loop system of robotic manipulator is robustfinite-time stable.

Remark 9 The proposed adaptive neural networkfinite-time controller is different from the con-

troller in [30], such as α is defined as 0 < α < 1while r = r1/r2 in [30], where r1 and r2 are positiveodd integers satisfying r2 > r1.

The block diagram of the overall closed-loopsystem is shown in Fig. 1.

5 Simulations and Experiments

In this section, the proposed adaptive neural net-work finite-time controller (NNFTC) is appliedto the control of a robot system to verify itseffectiveness.

To evaluate the performance of the proposedcontroller, the following performance criteria areused:

(1) Integral of the absolute value of the error(IAE)

I AE =∫ t f

0|e (t)| dt (48)

(2) Integral of the time multiplied by the ab-solute value of the error (ITAE)

IT AE =∫ t f

0t |e (t)| dt (49)

J Intell Robot Syst

Fig. 2 Two-link planarrobot

185 mm170 mm

Link 1

Link 2

(3) Integral of the square value (ISV) of thecontrol input

ISV =∫ t f

0u2 (t) dt (50)

Both IAE and ITAE are defined as the objec-tive numerical measures of tracking performancefor an entire error curve, where t f represents thetotal running time. The criterion IAE will givean intermediate result. In ITAE, time appears asa factor, which will heavily emphasize the errors

that occur late in time. The criterion ISV showsthe consumption of energy.

5.1 Simulations

Considering a two-link planar robotic manipula-tor shown in Fig. 2,

[m11 m12

m12 m22

] [ ..q1..q2

]+

[−2bq2, −bq2

bq1, 0

] [q1

q2

]

+[

f1

f2

]=

[τ1

τ2

](51)

0 0.5 1 1.5 2 2.5time (s)

3 3.5 4 4.5 5 0 0.5 1 1.5 2 2.5time (s)

(a) (b)

3 3.5 4 4.5 5-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

Pos

ition

err

or (

rad)

Joint 1

NNFTCFIDC

-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02Joint 2

NNFTCFIDC

Pos

ition

err

or (

rad)

Fig. 3 Trajectory tracking error

J Intell Robot Syst

0 0.5 1 1.5 2 2.5

time (s)3 3.5 4 4.5 5 0 0.5 1 1.5 2 2.5

time (s)

(a) (b)

3 3.5 4 4.5 5

Joint 1 Joint 2

-20

-15

-10

-5

0

5

10

15

20

Con

trol

torq

ue (

N m

)

NNFTCFIDC

-10

-8

-6

-4

-2

0

2

4

6

8

10NNFTCFIDC

Con

trol

torq

ue (

N m

)

Fig. 4 Control input torque

where

m11 = p1 + p2 + 2p3 cos q2 − 2p4 sin q2

m12 = p2 + p3 cos q2 − p4 sin q2

m22 = p2

b = p3 sin q2 + p4 cos q2

f1 = fv1q1 + fc1sgn (q1)

f2 = fv2q2 + fc2sgn (q2)

p1, p2, p3 and p4 are all the minimum inertia pa-rameters of robot, fv1, fc1 and fv2, fc2 are viscous,Coulomb friction coefficients of Joint1 and Joint2, respectively.

The actual parameters of robot are p1 =0.0398 kgm2, p2 = 0.0026 kgm2, p3 = −0.0015kgm2, p4 = 0.0081 kgm2, and fv1 = 0.534684,fc1 = 0.81958, fv2 = 0.001, fc2 = 0.002. The esti-mated parameters are p1 = 0.0286 kgm2, p2 =0.01624 kgm2, p3 = −0.00045 kgm2, p4 =0.004923 kgm2, and fv1 = 0.634384, fc1 = 0.91748,fv2 = 0.01196, fc2 = 0.044647. The referencesignals are given by qd1 = sin(2π t), qd2 =sin(2π t). The initial values of the system areselected as q1(0) = 0.2 rad and q2(0) = 0.2 rad.

The external disturbance τd = [0.2sin(10t),0.1sin(10t)]T . The control parameters are selectedas = 100, α = 0.8, K1 = diag(50,50), K2 =diag(50, 50) and � = 0.01.

In order to demonstrate the advantage of NN-FTC, a comparison with the finite-time inversetracking control (FIDC) proposed by Su et al. [13]is carried out. The FIDC law is given by

τ = M0 (q)[ ..q d − KpSig (e)α1 − KdSig

( .e)α2

]

+C0

(q,

.q) .

q +F0

( .q)

(52)

where Kp = diag(1000, 1000), Kd = diag(500,

500), α1 = 0.7, α2 = 2α1/(α1 + 1) = 0.8235 andF0 = [ f1, f2]T .

Figure 3 shows the trajectory tracking error ofthe two links, and Fig. 4 shows the control inputtorque of the two links. As shown in Figs. 3 and 4,NNFTC is of stronger robustness to dynamic un-certainties and external disturbances than FIDC.

Table 1 The performance indices

Controller Joint IAE ITAE ISV(rad) (rad-s) (104v2)

NNFTC Joint 1 4.3674 0.9415 1.958FIDC Joint 1 14.2281 8.9317 15.223NNFTC Joint 2 1.6020 1.3855 0.296FIDC Joint 2 4.5272 9.8730 4.447

J Intell Robot Syst

Fig. 5 Photo of thetwo-link planar robot forexperiments

AC servo motor

AC servo motor

EncoderJoint 2Joint 1

Moreover, it can be easily found that the trackingperformance of FIDC greatly deteriorates whenthere is no exact dynamic knowledge of robotmanipulators and when unmodeled dynamics aretaken into account. From the performance indiceslisted in Table 1, it is obvious that NNFTC notincreases but decreases the control input torqueas the tracking performance is improved. Thus,the performance and energy of control systemeffectively match. Furthermore, no “chattering”phenomenon occurs in the NNFTC. Thus, it isconcluded that the NNFTC is of high perfor-

mance, such as fast response, high tracking pre-cision and strong robustness.

5.2 Experiments

In order to further verify the effectiveness ofthe proposed NNFTC, some experiments werecarried out. In the experiments, the robot isdriven by two AC servo motors, as shown inFig. 5. Moreover, in order to ensure the real-timeperformance, the control algorithm is realizedin DSP Card, which also includes multi-channel

0 0.5 1 1.5 2 2.5

time (s)3 3.5 4 4.5 5 0 0.5 1 1.5 2 2.5

time (s)

(a) (b)

3 3.5 4 4.5 5

Joint 1 Joint 2

-1.5

-1

-0.5

0

0.5

1

1.5

Tra

ject

ory

(rad

)

DesiredActual

-1.5

-1

-0.5

0

0.5

1

1.5DesiredActual

Tra

ject

ory

(rad

)

Fig. 6 Trajectory tracking of robot

J Intell Robot Syst

0 0.5 1 1.5 2 2.5time (s)

3 3.5 4 4.5 5 0 0.5 1 1.5 2 2.5time (s)

(a) (b)

3 3.5 4 4.5 5

Joint 1 Joint 2

-0.025

-0.02

-0.015

-0.01

-0.005

0

0.005

Tra

ckin

g er

ror

(rad

)

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

Tra

ckin

g er

ror

(rad

)

Fig. 7 Tracking error of robot

digital/analog(D/A) and encoder interface cir-cuits, and the control period is set as 2 ms. Uncer-tain factors are given to verify the robot system’srobustness to dynamic parameter uncertaintiesand joint frictions. The desired trajectory is qd =[sinπ t, sinπ t]T , and the control parameters are thesame as those in the simulations.

Figures 6, 7 and 8 respectively show the ex-perimental results of trajectory tracking, trackingerror and control input torque. It can be seen thatthe tracking error is less than 0.005 rad and the re-sponse time is less than 0.2 s, which demonstratesthat the NNFTC system is of satisfying controlperformance.

0 0.5 1 1.5 2 2.5time (s)

3 3.5 4 4.5 5 0 0.5 1 1.5 2 2.5time (s)

(a) (b)

3 3.5 4 4.5 5

Joint 1 Joint 2

-4

-2

0

2

4

6

8

10

12

Tor

que

(Nm

)

Tor

que

(Nm

)

-1.5

-1

-0.5

0

0.5

1

1.5

Fig. 8 Control input torque of robot

J Intell Robot Syst

6 Conclusions

Proposed in this paper is an adaptive neural net-work finite-time controller for a class of uncer-tain nonlinear systems, which is then applied toan uncertain robotic manipulators. The controllerenables the finite-time convergence of the systemstate, and possesses satisfying performances suchas fast response, high precision and strong robust-ness. Simulated and experimental results verifythe effectiveness of the proposed controller. Theproposed method can also be applied to otheruncertain dynamic systems, such as DC motorsystems. Our future work is to design a robustvelocity observer, which is more applicable torobotic systems.

Acknowledgements The authors are most grateful tothe anonymous reviewers for their constructive commentswhich have helped to improve the quality of this paper.The work is supported by the National High-TechnologyResearch and Development Program (“863” Key Pro-gram 2009AA043901-3), the Strategic Emerging Indus-try Projects of Guangdong Province (2011A091101001),the Scientific and Technological Projects of GuangdongProvince (2012B010900076, 2012B011300056), and the Sci-ence and Technology Planning Project of Zhanjiang City(2012A0103).

Appendix

Appendix A

When∣∣ zδ

∣∣ ≥ 1, sat (z) = sgn( z

δ

), the proof can be

readily accomplished by following the proof ofTheorem 3.1, and thus being omitted here.

Inside the boundary layer, once sat (z) = zδ, the

first-order derivative of V2 is given by.

V 2 = xT1

.x 1 + zT .

z

= −xT1 K1Sig (x1)

α − zT K1Sig (z)α

−zT (K3sat (z) − w)

= −xT1 K1Sig (x1)

α − zT K1Sig (z)α

−zT(

K3zδ

− w)

≤ −xT1 K1Sig (x1)

α − zT K1Sig (z)α − ‖z‖×

(k3

‖z‖δ

− d)

where k3 = min {k31, k32, ..., k3n}.

If δ is small enough, i.e. δ ≤ k3d ‖z‖, one can

obtain

.

V 2 = xT1

.x1+zT .

z

≤ −xT1 K1Sig (x1)

α − zT K1Sig (z)α

− ‖z‖(

k3‖z‖δ

− d)

≤ −xT1 K1Sig (x1)

α − zT K1Sig (z)α

≤ −k1Vμ

1 − k2

(12

n∑i=1

z2i

)μ

≤ −kVμ

2

The proof is completed.

References

1. Bhat, S.P., Bernstein, D.S.: Lyapunov analysis of finite-time differential equations. In: Proceedings of theAmerican Control Conference 1995, vol. 1833, pp.1831–1832 (1995). 21–23 Jun 1995

2. Bhat, S.P., Bernstein, D.S.: Finite-time stability of ho-mogeneous systems. In: Proceedings of the AmericanControl Conference 1997, vol. 2514, pp. 2513–2514(1997). 4–6 Jun 1997

3. Bhat, S.P., Bernstein, D.S.: Continuous finite-time sta-bilization of the translational and rotational double in-tegrators. IEEE Trans. Autom. Control 43(5), 678–682(1998)

4. Bhat, S.P., Bernstein, D.S.: Finite-time stability of con-tinuous autonomous systems. SIAM J. Control Optim.38(3), 751–766 (2000)

5. Bhat, S.P., Bernstein, D.S.: Geometric homogeneitywith applications to finite-time stability. Math. Con-trol Signal 17(2), 101–127 (2005). doi:10.1007/s00498-005-0151-x

6. Yiguang, H., Zhong-Ping, J.: Finite-time stabilizationof nonlinear systems with parametric and dynamic un-certainties. IEEE Trans. Autom. Control 51(12), 1950–1956 (2006)

7. Zhang, X., Feng, G., Sun, Y.: Finite-time stabilizationby state feedback control for a class of time-varyingnonlinear systems. Automatica 48(3), 499–504 (2012).doi:10.1016/j.automatica.2011.07.014

8. Huang, X., Lin, W., Yang, B.: Global finite-timestabilization of a class of uncertain nonlinear sys-tems. Automatica 41(5), 881–888 (2005). doi:10.1016/j.automatica.2004.11.036

9. Ji, L., Chunjiang, Q.: Global finite-time stabilizationby dynamic output feedback for a class of continuous

J Intell Robot Syst

nonlinear systems. IEEE Trans. Autom. Control 51(5),879–884 (2006)

10. Hong, Y., Xu, Y., Huang, J.: Finite-time control forrobot manipulators. Syst. Control Lett. 46(4), 243–253(2002)

11. Su, Y.: Global continuous finite-time tracking of robotmanipulators. Int. J. Robust Nonlinear Cont. 19(17),1871–1885 (2009). doi:10.1002/rnc.1406

12. Zhao, D., Li, S., Zhu, Q., Gao, F.: Robust finite-timecontrol approach for robotic manipulators. IET Cont.Theor. Appl. 4(1), 1–15 (2010)

13. Su, Y., Zheng, C.: Global finite-time inverse track-ing control of robot manipulators. Robot. Comput. In-tegr. Manuf. 27(3), 550–557 (2011). doi:10.1016/j.rcim.2010.09.010

14. de Jesus Rubio, J., Torres, C., Aguilar, C.: Optimal con-trol based in a mathematical model applied to roboticarms. Int. J. Innov. Comp. Inform. Control 7(8), 5045–5062 (2011)

15. Torres, C., Jesús Rubio, J., Aguilar-Ibáñez, C., Pérez-Cruz, J.H.: Stable optimal control applied to a cylin-drical robotic arm. Neural Comput. Appl. 1–8 (2012).doi:10.1007/s00521-012-1294-6

16. Liu, H., Zhang, T.: Fuzzy sliding mode control of ro-botic manipulators with kinematic and dynamic un-certainties. J. Dyn. Syst. Meas. Control 134(6), 061007(2012)

17. Yu, S., Yu, X., Shirinzadeh, B., Man, Z.: Continu-ous finite-time control for robotic manipulators withterminal sliding mode. Automatica 41(11), 1957–1964(2005). doi:10.1016/j.automatica.2005.07.001

18. Jesus Rubio, J., Soriano, L.A.: An asymptotic stableproportional derivative control with sliding mode grav-ity compensation and with a high gain observer for ro-botic arms. Int. J. Innov. Comp. Inform. Control 6(10),4513–4526 (2010)

19. de Jesus Rubio, J.: SOFMLS: online self-organizingfuzzy modified least-squares network. IEEE Trans.Fuzzy Syst. 17(6), 1296–1309 (2009). doi:10.1109/tfuzz.2009.2029569

20. Chen, C.-W.: Stability analysis and robustness design ofnonlinear systems: an NN-based approach. Appl. SoftComput. 11(2), 2735–2742 (2011) doi:10.1016/j.asoc.2010.11.004

21. Yeh, K., Chen, C.-W., Lo, D., Liu, K.F.: Neural-network fuzzy control for chaotic tuned mass dampersystems with time delays. J. Vib. Control 18(6), 785–795 (2012). doi:10.1177/1077546311407538

22. Hsu, C.-F., Lin, C.-M., Yeh, R.-G.: Supervisory adap-tive dynamic RBF-based neural-fuzzy control systemdesign for unknown nonlinear systems. Appl. Softw.

Comput. 13(4), 1620–1626 (2013). doi:10.1016/j.asoc.2012.12.028

23. Perez-Cruz, J.H., Rubio, J.d.J., Ruiz-Velzquez, E.,Solis-Perales, G.: Tracking control based on recurrentneural networks for nonlinear systems with multiple in-puts and unknown deadzone. Abstr. Appl. Anal. 2012,18. (2012). doi:10.1155/2012/471281

24. Pérez-Cruz, J.H., Ruiz-Velázquez, E., Rubio, J.D.J.,Padilla, C.A.D.A.: Robust adaptive neurocontrol ofsiso nonlinear systems preceded by unknown dead-zone. Math. Probl. Eng. 2012, 23 (2012). doi:10.1155/2012/342739

25. Huang, S.N., Tan, K.K., Lee, T.H.: Adaptive neuralnetwork algorithm for control design of rigid-link elec-trically driven robots. Neurocomputing 71(4–6), 885–894 (2008). doi:10.1016/j.neucom.2007.02.012

26. Peng, J., Wang, J., Wang, Y.: Neural network basedrobust hybrid control for robotic system: an Hinfapproach. Nonlinear Dyn. 65(4), 421–431 (2011).doi:10.1007/s11071-010-9902-4

27. Sun, T., Pei, H., Pan, Y., Zhou, H., Zhang, C.: Neuralnetwork-based sliding mode adaptive control for robotmanipulators. Neurocomputing 74(14–15), 2377–2384(2011). doi:10.1016/j.neucom.2011.03.015

28. Zuo, Y., Wang, Y., Liu, X., Yang, S.X., Huang, L.,Wu, X., Wang, Z.: Neural network robust H∞ trackingcontrol strategy for robot manipulators. Appl. Math.Model. 34(7), 1823–1838 (2010)

29. Rubio, J.: Modified optimal control with a back prop-agation network for robotic arms. IET Control TheoryAppl. 6(14), 2216–2225 (2012)

30. Wang, L.Y., Chai, T.Y., Zhai, L.F.: Neural-network-based terminal sliding-mode control of robotic ma-nipulators including actuator dynamics. IEEE Trans.Ind. Electron. 56(9), 3296–3304 (2009). doi:10.1109/tie.2008.2011350

31. Liu, H., Zhang, T.: Neural network-based robustfinite-time control for robotic manipulators consid-ering actuator dynamics. Robot. Comput. Integr.Manuf. 29(2), 301–308 (2013). doi:10.1016/j.rcim.2012.09.002

32. Haimo, V.T.: Finite time controllers. SIAM J. Control.Optim. 24(4), 760–770 (1986). doi:10.1137/0324047

33. De Jesús Rubio, J., Yu, W.: A new discrete-timesliding-mode control with time-varying gain and neuralidentification. Int. J. Control 79(4), 338–348 (2006).doi:10.1080/00207170600566188

34. Rodríguez, F., Jesús Rubio, J., Gaspar, C.M., Tovar,J., Armendáriz, M.M.: Hierarchical fuzzy CMAC con-trol for nonlinear systems. Neural Comput. Appl. 1–9(2013). doi:10.1007/s00521-013-1423-x