Nonlinear DynDOI 10.1007/s11071-014-1477-z

ORIGINAL PAPER

Fuzzy adaptive output feedback control for robotic systemsbased on fuzzy adaptive observer

Jinzhu Peng · Yan Liu · Jie Wang

Received: 10 October 2013 / Accepted: 22 May 2014© Springer Science+Business Media Dordrecht 2014

Abstract In this paper, a fuzzy adaptive output feed-back control scheme based on fuzzy adaptive observeris proposed to control robotic systems with parameteruncertainties and external disturbances. It is supposedthat only the joint positions of the robotic system can bemeasured, whereas the joint velocities are unknown andunmeasured. First, a fuzzy adaptive nonlinear observeris presented to estimate the joint velocities of roboticsystems, and the observation errors are analyzed usingstrictly positive real approach and Lyapunov stabilitytheory. Next, based on the observed joint velocities, afuzzy adaptive output feedback controller is developedto guarantee stability of closed-loop system and achievea certain tracking performance. Based on the Lyapunovstability theorem, it is proved that all the signals inclosed-loop system are bounded. Finally, simulationexamples on a two-link robotic manipulator are pre-sented to show the efficiency of the proposed method.

Keywords Fuzzy logic systems · Output feedbackcontrol · State observer · Robotic system

J. Peng (B) · J. WangSchool of Electrical Engineering, Zhengzhou University,Zhengzhou 450001, Henan, People’s Republic of Chinae-mail: jzpeng@zzu.edu.cn

Y. LiuLibrary of Zhengzhou University, Zhengzhou University,Zhengzhou 450001, Henan, People’s Republic of China

1 Introduction

Due to the uncertainties, disturbances and nonlinearsystem dynamics, tracking control for robotic systemsalways is a challenging problem and has been given alot of attention in the control field [1,2]. Many pow-erful methodologies have been applied on robotic sys-tems to achieve good tracking performances, such ascomputed torque method (CTC) [3–5], variable struc-ture system (VSS) [6,7], robust control [8–10], neuralnetworks (NN) [11–13], fuzzy logic system (FLS)[14–16], and among others.

To obtain a certain performance, distinct hybrid con-trol approaches have been designed to control roboticsystems using different combinations of the abovemethods. In such schemes, on one hand, an adaptiveFLS or NN system is usually employed to approxi-mate the uncertain nonlinearities of robotic systems,and then cope with approximation errors and externaldisturbances. These adaptive fuzzy or NN controllersare augmented by a robust compensator that can be asupervisory control [17] type, variable structure con-trol [18,19], and/or H∞ control [17,20]. On the otherhand, to utilize the nominal model of robotic systems,CTC plus an intelligent hybrid compensator were alsoproposed. Song et al. [3] proposed an approach of CTCcombined fuzzy systems, in which the nominal systemis controlled by CTC and for uncertain system, a fuzzycontroller acts as a compensator. Zuo et al. [4] used aneural network robust controller acting as compensator

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and assumed that the external disturbance has finiteenergy. Peng et al. [5] proposed a neural network-basedrobust hybrid tracking control scheme which combinesCTC, neural network, VSC, and H∞ control for roboticsystems.

The above researches generally assumed that thejoint positions and velocities could be measured accu-rately. However, in practical robotic systems, the jointpositions are measured by an encoder that gives accu-rate measurements. The joint velocities are usuallymeasured by velocity tachometers, which are expensiveand often contaminated by noises [21]. Therefore, it isimportant to develop effective observers to estimate thejoint velocities accurately. Freidovich and Khalil [22]proposed a high gain observer for uniformly observablenonlinear systems. The high gain observer is theoret-ically well founded and is able to offer the necessarydesign calculations analytically in applications. How-ever, because of the requirement of some assumptions,this type of observer could only be effective in a certainclass of nonlinear systems. Mosayebi et al. [23] alsoproposed a nonlinear high gain observer to estimatethe elastic degrees of freedom and their time deriv-atives. It should be pointed that most of the observergains would be chosen with very high values. However,high values of observer gain increase the sensitivity tonoise, and it could reduce the merit of practical appli-cation, therefore. FLS is a knowledge-based designmethodology, it has been widely used in system iden-tification, modeling, and controller design due to itsuniversal approximation ability [16,21,24–28]. Goléaet al. [25] developed a fuzzy adaptive robot controlbased on a linear state observer. Liu and Tong [26,27]proposed adaptive fuzzy tracking control for a classof uncertain nonlinear multiple-input–multiple-outputs(MIMO) systems based on an adaptive fuzzy observer.Mostefai et al. [24] proposed fuzzy observer-basedcontrol strategy for tracking robot system trajectorieswhen operating in zero velocity regions and duringmotion reversals. In addition, NN and fuzzy NN arealso applied to design the observer and/or controller ofrobotic manipulator. Wai et al. [29] designed an adap-tive fuzzy neural network velocity sensorless controlscheme for an n-link robot manipulator. Lin [30] pre-sented an H∞ fuzzy output feedback tracking controlscheme for robotic manipulators without measuringjoint velocities, where a fuzzy basis function network-based observer was developed to estimate joint veloci-ties. In this paper, only the joint positions are assumed

to be measured, while the joint velocities are obtainedby a fuzzy adaptive observer and the observation errorsare analyzed using strictly positive real approach andLyapunov stability theory. Then, based on the estimatedstates, a fuzzy adaptive output feedback control schemeis developed to control the robotic systems, and basedon Lyapunov stability theorem, the proposed controllercan guarantee stability of closed-loop systems and tra-jectory tracking performances. Finally, simulations areconducted on two-link robotic manipulator to test theperformance of the proposed control scheme.

This paper is organized as follows. In Sect. 2, sometheoretical preliminaries are addressed, which consistof mathematical notations, FLS, and details on dynam-ics of robotic system. In Sect. 3, the design of fuzzyadaptive observer is given, and its stability is analyzed.Based on the estimated states, the design of fuzzy adap-tive output feedback control scheme and its stabilityanalysis is given in Sect. 4. Simulation results are pre-sented to confirm the effectiveness of the proposedmodel in Sect. 5, and conclusions are drawn in Sect. 6.

2 Problem statement

In this paper, standard notations are used. Let � bethe real number set, �n be the n-dimensional vectorspace, �n×n be the n × n real matrix space. The normof vector x ∈ �n and that of matrix A ∈ �n×n aredefined as ‖x‖ = √

xTx and ‖A‖ = tr(AT A), respec-tively. If y is a scalar, then ‖y‖ denotes the absolutevalue. λmin(A) and λmax(A) are the minimum and themaximum eigenvalue of matrix A, respectively. In×n

is n × n identity matrix. sgn(·) is a standard signfunction. Denote ‖x‖2

A = xT Ax ≥ λmin(A)‖x‖2, Ais a positive symmetry matrix. The following defini-tion of strictly positive real (SPR) [31] and Kalman-Yakubovich-Popov (KYP) Lemma [21] is also statedin its generality.

Definition 1 The transfer matrix H(s) is said to bepositive real if: (i) all the elements of H(s) are analyticin Re[s] > 0, and (ii) H(s) + HT(−s) ≥ 0 for allRe[s] > 0. Furthermore, H(s) is said to be strictlypositive real (SPR) if H(s −ε) is positive real for someε > 0.

Lemma 1 Let H(s) = C(s I − A)−1 B be an n × ntransfer matrix such that H(s) + HT(−s) has normalrank n, where A is Hurwitz, (A, B) is stabilizable, and

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Fig. 1 The basic configuration of a FLS

(C, A) is observable. Then, H(s) is SPR if and only ifthere exist positive definite symmetric matrices P andQ, such that{

P A + AT P = −QP B = CT.

2.1 Description of FLS

A FLS comprises four parts as shown in Fig. 1, whichare the fuzzy rule base, the fuzzifier, the fuzzy inferenceengine working on fuzzy rules, and the defuzzifier. Thefuzzy inference engine performs a mapping from fuzzysets in input space U ∈ �n to fuzzy sets in output spaceV ∈ �m based on fuzzy rules, where

U = U1 × U2 × · · · × Un, Ui ∈ �for i = 1, 2, . . . , n

V = V1 × V2 × · · · × Vm, Vj ∈ �for j = 1, 2, . . . , m

Generally, if an MIMO system is considered with ninputs x = [x1, x2, . . . , xn]T, and the output data y =[y1, y2, . . . , ym]T, the i th fuzzy rule has the followingform, Ri : If x1 is Ai

1 and …xn is Ain then y1 is Bi

1 and…ym is Bi

m, i = 1, 2, . . . , M , where i is the number offuzzy rules, M is the total number of rules, Ai

p and Biq

(p = 1, . . . , n, q = 1, . . . , m) are the fuzzy sets of theantecedent part and the real numbers of the consequentpart, described by their membership functions μAi

p(x p)

and μBiq(yq), respectively.

When the inputs x = [x1, x2, . . . , xn]T are given,the output y of the fuzzy inference system withweighted-center defuzzifier, product inference, and sin-gleton fuzzifier can be derived from the following form,

y(x) =∑m

i=1 yi (∏n

p=1 μAip(x p))∑m

i=1 (∏n

p=1 μAip(x p))

, (1)

where yi is the point in Vj at which μBiq(yi ) achieves its

maximum value. Without loss of generality, we assumethat μBi

q(yi ) = 1.

Assume that fuzzy basis functions are defined asfollows:

φi (x) =∏n

p=1 μAip(x p)∑m

i=1(∏n

p=1 μAip(x p))

(2)

then, the fuzzy system in Eq. (1) can be rewritten as

y(x) =m∑

i=1

yiφi (x) = θTφ(x), (3)

where φ(x) = [φ1(x), φ2(x), . . . , φm(x)]T ∈ �M iscalled the fuzzy basic function vector or the antecedentfunction vector, and θ = [y1, y2, . . . , ym]T ∈ �M iscalled the parameter vector.

It has been proven that the FLS can approximate anyreal continuous function over a compact set to arbitraryaccuracy [16].

2.2 Robotic manipulator dynamics and properties

The dynamic equations of robotic systems are usuallyobtained by the Euler–Lagrange equations. Considera general n-link rigid robot, which takes into accountthe external disturbances, with the equation of motiongiven by [2,5]

M(q)q + C(q, q)q + G(q) + τd = τ, (4)

where q, q , and q ∈ �n are joint angular position vec-tors, velocity vectors, and acceleration vectors of therobot, respectively; M(q) ∈ �n×n is the symmetric andpositive definite inertia matrix; C(q, q) ∈ �n×n is theeffect of Coriolis and centrifugal forces; G(q) ∈ �n isthe gravity vector; τd ∈ �n denotes an unknown distur-bances including unstructured dynamics and unknownpayload dynamics. τ ∈ �n is the torque input vec-tor.

The following properties are required for the subse-quent development.

Property 1 The inertia matrix M(q) is symmetric andpositive definite, which is uniformly bounded and sat-isfies:

mm In×n ≤ M(q) ≤ mM In×n, ∀q ∈ �n,

where mm and mM are some positive constants.

Property 2 The matrix (M(q) − 2C(q, q)) is skew-symmetric, i.e.,

xT(M(q) − 2C(q, q))x = 0, ∀x ∈ �n .

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Property 3 The norm C(q, q) is bounded and satis-fies:

‖C(q, q)‖ ≤ Cb‖q‖,where Cb is a positive constant.

Assumption 1 Disturbance is bounded by ‖τd‖ ≤ τD,where τD is some positive constant.

The parameters M(q), C(q, q), and G(q) in dynam-ical model Eq. (4) are functions of physical parame-ters of manipulators like links masses, links lengths,moments of inertial, and so on. The precise valuesof these parameters are difficult to acquire due tomeasuring errors, environment, and payloads varia-tions. Therefore, here it is assumed that actual valuesM(q), C(q, q), and G(q) can be separated as nom-inal parts denoted by M0(q), C0(q, q), and G0(q),and uncertain parts denoted by �M(q),�C(q, q), and�G(q), respectively. These variables satisfy the fol-lowing relationships:

M(q) = M0(q) + �M(q)

C(q, q) = C0(q, q) + �C(q, q)

G(q) = G0(q) + �G(q).

Assumption 2 The bounds of uncertainty parametersare known, which can be expressed as

‖�M(q)‖ ≤ δM , ‖�C(q, q)‖ ≤ δC , ‖�G(q)‖ ≤ δG,

where δM , δC , and δG are positive constants, and δM <

λmin(M0(q)).

3 Fuzzy adaptive observer design and stabilityanalysis

Let x1 = q and x2 = q , the robotic system dynamicequation (4) can be written as{

x1 = x2

x2 = M−1(x1)(τ − τd) − Ho(x1, x2),(5)

where

Ho(x1, x2) = M−1(x1) [C(x1, x2)x2 + G(x1)] . (6)

In this paper, the unknown function Ho(x1, x2) isapproximated by using FLS, i.e.,

Ho(x1, x2) = θ∗To φo(x1, x2) + εo(x1, x2), (7)

where φo(x1, x2) denotes generalization result, θo is theweight matrix, εo(x1, x2) is a minimum reconstructed

error vector. We assume that there exists θo , and idealparameter is in the compact set θo , which is definedas θo = {θo ∈ �m×n : ‖θo‖ ≤ Eθo}. The ideal para-meter can be defined as

θ∗o = arg min

θo∈θo

{sup

∣∣∣Ho(x1, x2) − θTo φo(x1, x2)

∣∣∣} .

(8)

3.1 Fuzzy adaptive observer design

A nonlinear state observer based on FLS can bedesigned as follows:⎧⎨⎩

˙z1 = x2 + K1 x1˙z2 = −θTo φo(x1, x2) + K2 x1 + M−1

0 (x1)τ

−υo(x1, x2),

(9)

where x1 and x2 are the observed values of state vari-ables x1 and x2, respectively, the gain K1 and K2 areboth positive matrix, and υo(x1, x2) is the robust term,which is used to compensate the external disturbancesand approximation error of FLS.

The observer error can be defined as

x1 = x1 − x1, x2 = x2 − x2. (10)

Then, the observed states x1 and x2 can be expressedas{

x1 = z1

x2 = z2 + K3 x1,(11)

where K3 = K T3 > 0 is the gain matrix.

Taking the derivative of Eq. (11) with respect totime and considering Eq. (9), the dynamic equation ofobserver can be written as⎧⎨⎩

˙x1 = x2 + K1 x1˙x2 = −θTo φo(x1, x2) + K2 x1 + K3 ˙x1

+M−10 (x1)τ − υo(x1, x2).

(12)

Substituting Eq. (12) to Eq. (10), the dynamic errorequation of observer can be obtained⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

˙x1 = −K1 x1 + x2˙x2 = M−1(x1)(τ − τd) − M−1(x1)

× [C(x1, x2)x2 + G(x1)]

−[−θT

o φo(x1, x2) + K2 x1 + K3 ˙x1

+M−10 (x1)τ − υo(x1, x2)

].

(13)

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Furthermore, the dynamic error equation of observerEq. (13) can be rewritten as⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

˙x1 = −K1 x1 + x2˙x2 = −M−1(x1)�M(x1)M−10 (x1)τ

−K2 x1 − K3 ˙x1 − υo(x1, x2)

−M−1(x1)τd + {θT

o φo(x1, x2)

−M−1(x1) [C(x1, x2)x2 + G(x1)]}.

(14)

Assume that x2 can not be measured, then systemuncertainties have to be approximated by using x1 andx2, therefore, we have

θTo φo(x1, x2) − M−1(x1) [C(x1, x2)x2 + G(x1)]

= θTo φo(x1, x2) − θ∗T

o φo(x1, x2) − εo(x1, x2)

= θTo φo(x1, x2) − θ∗T

o

[φo(x1, x2) − φo(x1, x2)

]−εo(x1, x2)

= θTo φo(x1, x2) − θ∗T

o φo(x1, x2, x2) − εo(x1, x2).

(15)

where θo = θo − θ∗o is the adjustable parameter

estimated error and φo(x1, x2, x2) = φo(x1, x2) −φo(x1, x2) is the fuzzy basis function vector estimatederror.

Let

ωo(x1, x2) = θ∗To φo(x1, x2, x2)

= θ∗To

[φo(x1, x2) − φo(x1, x2)

]. (16)

Assumption 3 The fuzzy basis function vector esti-mated error and reconstructive error are bounded, andsatisfy

‖ωo(x1, x2)‖ ≤ Eθoδφo , ‖εo(x1, x2)‖ ≤ δε.

Remark 1 The fuzzy basis function vector is bounded,i.e., ‖φo(x1, x2)‖ is bounded, then estimated error‖φo(x1, x2, x2)‖ is bounded, let ‖φo(x1, x2, x2)‖ ≤δφo . Furthermore, θ∗

o is also bounded, i.e., ‖θ∗o ‖ ≤ Eθo .

Therefore, the bound ‖ωo(x1, x2)‖ in Assumption 3 isreasonable.

Substituting Eqs. (15) and (16) into Eq. (14), theobserver error dynamic equation can be rewritten as⎧⎪⎪⎨⎪⎪⎩

˙x1 = −K1 x1 + x2˙x2 = (K3 K1−K2)x1−K3 x2−M−1(x1)�M(x1)

×M−10 (x1)τ −M−1

0 (x1)τd +θTo φo(x1, x2)

−ω(x1, x2) − εo(x1, x2) + υo(x1, x2).

(17)

Equation (17) can be expressed by the state equationas follows:

⎧⎨⎩

˙x = Aox + Bo{υo(x1, x2) − M−1(x1)�M(x1)

×M−10 (x1)τ − M−1

0 (x1)τd + θTo φo(x1, x2)

− ω(x1, x2) − εo(x1, x2)} x1 = Cox,

,

(18)

where

Ao =[ −K1 In×n

K3 K1 − K2 −K3

],

Bo =[

0n×n

In×n

], Co = [

In×n 0n×n].

According to Definition 1 of SPR, we define theoutput of error dynamic equation (18) as

x1 = H(s){υo(x1, x2)−M−1(x1)�M(x1)M−1

0 (x1)τ

−M−10 (x1)τd +θT

o φo(x1, x2)−ω(x1, x2), (19)

−εo(x1, x2)}

,

where H(s) = Co(s I − Ao)−1 Bo is a stable transfer

function. The matrices K1, K2, and K3 can be properlychosen to guarantee Ao is a Hurwitz matrix and H(s)is SPR.

Consider the inequations as follows:

‖M−10 (x1)‖ ≤ 1

λmin(M0(x1)) − δM(20)

‖M−1(x1)�M(x1)‖ ≤ δM

λmin(M0(x1)) − δM. (21)

Then, the term including the external disturbance andmodeling error of fuzzy system is bounded as

‖ − M−10 (x1)τd − ωo(x1, x2) − εo(x1, x2)‖

≤ τD

λmin(M0(x1)) − δM+ Eθoδφo + δε � ρo. (22)

3.2 Stability analysis

Theorem 1 Considering the robotic system dynamicequation (5), suppose that Assumptions 1–3 are satis-fied, and the transfer function H(s) is SPR, accordingto Lemma 1, if there exists a matrix P = PT > 0 suchthat the following matrix equality holds{

ATo P + P Ao − P Bo R−1 BT

o P = −QP Bo = CT

o .(23)

Then, a nonlinear observer based on FLS can bedefined as Eq. (9), where the robust term can be definedas follows

υo(x1, x2) = υo1(x1, x2) + υo2(x1, x2), (24)

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where

υo1(x1, x2) = −R−1 BTo P x =−R−1Cox = −R−1 x1

(25)

υo2(x1, x2)=⎧⎨⎩

−[

δM ‖M−10 (x1)τ‖

λmin(M0(x1))−δM+ρo

]x1‖x1‖ , if x1 �= 0

0, if x1 = 0

(26)

The projection algorithm is adopt as the following fuzzyadaptive law

˙θo =

⎧⎪⎪⎪⎨⎪⎪⎪⎩

−Γ −1o φo(x1, x2)xT

1 , If(‖θo‖ < Eθo

)or(

‖θo‖ = Eθo and xT1 θT

o φo(x1, x2) ≥ 0)

Pr(·), if(‖θo‖ = Eθo and xT

1 θTo φo(x1, x2) < 0

)

(27)

Pr(·) = Γ −1o φo(x1, x2)xT

1 − Γ −1o xT

1 θTo φo(x1, x2)φo(x1, x2)

‖θo‖θo

(28)

where Γo is a positive matrix. The observer error x andadaptive parameters error θo are bounded.

Proof Choose a Lyapunov function candidate as

Vo(x, θo, t) = 1

2xT Px + 1

2tr{θT

o Γoθo}. (29)

Taking the derivative of the Lyapunov function withrespect to time and substituting Eqs. (18) and (23),yields

Vo(x, θo, t)= 1

2˙xT Px + 1

2xT P ˙x + tr{θT

o Γo˙θo}

= 1

2xT(

ATo P+P Ao

)x +1

2

{υo(x1, x2)

− M−1(x1)�M(x1)M−10 (x1)τ

− M−1(x1)τd+θTo φo(x1, x2)−ω(x1, x2)

− εo(x1, x2)}T BTo P x + 1

2xT P Bo

×{υo(x1, x2) − M−1(x1)�M(x1)

× M−10 (x1)τ − M−1(x1)τd

+ θTo φo(x1, x2) −ω(x1, x2)

− εo(x1, x2)}

+ tr{θTo Γo

˙θo}.

(30)

According to inequality (22) and Eq. (26), we canobtain{

υo2(x1, x2) − M−1(x1)�M(x1)M−10 (x1)τ

−M−1(x1)τd −ω(x1, x2)−εo(x1, x2)}T

BTo P x ≤0.

(31)

Since BTo P x = Cox = x1, and considering the projec-

tion algorithm Eqs. (27) and (28), we can obtain{θT

o φo(x1, x2)}T

BTo P x + tr{θT

o Γo˙θo} ≤ 0 (32)

Considering Eq. (25), and substituting Eqs. (31) and(32) into Eq. (30), yields

Vo(x, θo, t) ≤ −1

2xT Qx + xT P Bo R−1 BT

o P x

+ xT P Boυo1 = −1

2xT Qx . (33)

Integrating the inequality (33) from t = 0 to t = Tyields

Vo(x(t), θo(t)) − Vo(x(0), θo(0)) ≤ −1

2

t∫0

xT Qxdτ

(34)

Since

Vo

(x(0), θo(0)

)= 1

2xT(0)Px(0)

+1

2tr{θT

o (0)Γoθo(0)}

(35)

the inequality (34) leads to

‖x‖ ≤

√√√√ xT(0)Px(0) + 12 tr{θT

o (0)Γoθo(0)}

λmin(Q). (36)

It can be concluded that the observer error x and adap-tive parameters error θo are bounded. �

4 Fuzzy adaptive output feedback controllerdesign and stability analysis

In this section, we will design a fuzzy adaptive outputfeedback control (FAOFC) law based on the measuredjoint positions and observed joint velocities such thatjoint motions of robotic systems Eq. (4) can follow thedesired trajectories and all signals of the closed-loopsystem are bounded.

4.1 Output feedback controller design

Assume that the states of robotic systems Eq. (4) are allmeasurable, that is, q and q are measurable, if the uncer-tainties and disturbances are ignored, i.e., f (q, q) = 0

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and τd = 0, the robotic system model Eq. (4) can beconverted into the following nominal model

M0(q)q + C0(q, q)q + G0(q) = τ. (37)

The corresponding output feedback control law forthe nominal model Eq. (37) of robotic system can bechosen as

τ0 = M0(q)(qd+Kv e+K pe

)+C0(q, q)q + G0(q),

(38)

where e is the tracking error defined by e = q − qd, q,and qd are the actual and desired joint trajectories,respectively. The coefficients Kv and K p are the gainmatrices, which should be chosen such that all the rootsof the polynomial h(s) = s2 + Kvs + K p are in theopen left-half plane.

Assumption 4 The desired trajectories qd are con-tinuous and bounded known functions of time withbounded known derivatives up to the second order.

Substituting Eq. (38) into Eq. (37) yields

e + Kv e + K pe = 0. (39)

Obviously, the output feedback control techniqueresults in a linear time-invariant closed-loop systemEq. (39), implying acquirement of globally asymptoti-cal stability. Furthermore, globally asymptotical stabil-ity is guaranteed provided that Kv and K p in Eq. (38)are symmetric and positive definite constant matrices.

According to the above analysis, the output feed-back control strategy, which is also called as computedtorque control method, relies on strong assumptionsthat exact knowledge of robotic dynamics is preciselyknown and unmodeled dynamics has to be ignored.However, these assumptions are impossible in practi-cal engineering. Furthermore, to design the output feed-back control law Eq. (38), the full states (q and q) haveto be measurable.

4.2 Fuzzy adaptive output feedback controller design

In this section, we assume that only the joint positionsare measurable, while the joint velocities can not bemeasured directly. The joint velocities can be obtainedby the stable fuzzy adaptive observer in Sect. 3.

Let the states be x1 = q and x2 = q , assume thatthe desired trajectories are xd = (

qTd , qT

d

)T, then the

tracking errors can be defined as

e = qd − x1, e = qd − x2. (40)

Since the joint velocities cannot be known directly,the stable fuzzy adaptive observer in Sect. 3 is used toacquire the estimated velocity values x2 = x2 − x2,where x2 are the errors between the estimated velocityvalues and actual velocity values. Then, the trackingerrors can be modified as,

e = qd − x1, ˙e = qd − x2. (41)

Define a filtered tracking error as

r = ˙e + �e, (42)

where � = �T > 0 is the positive matrix.Differentiating Eq. (42) and multiply M0(x1) on

both sides, we can obtain

M0(x1) ˙r = M0(x1) ¨e + M0(x1)�e

= M0(x1)(

qd + ˙x2 + �e)

− M0(x1)x2

= M0(x1)(

qd + ˙x2. + �e)

(43)

− [τ − C0(x1, x2)x2 − G0(x1)

− f (x1, x2) − τd ]

= −C0(x1, x2)r − τ + τd + Hc(x1, x2)

where f (x1, x2) = �M(x1)(qd + ˙x2) + �C(x1, x2)

qd + �G(x1), and define

Hc(x1, x2) = M0(x1)(

qd + ˙x2 + �e)

+ C0(x1, x2) (qd + x2 + �e)

+ G0(x1) + f (x1, x2). (44)

In this paper, the unknown function Hc(x1, x2) isapproximated using FLS, i.e.,

Hc(x1, x2) = θ∗Tc φc(x1, x2) + εc(x1, x2), (45)

where φc(x1, x2) denotes generalization result, θo isthe weight matrix, and εc(x1, x2) is a minimum recon-structed error vector. We assume that there exists θc ,and ideal parameter is in the compact set θc , whichis defined as θc = {θc ∈ �m×n : ‖θc‖ ≤ Eθc }. Theideal parameter can be defined as

θ∗c = arg min

θc∈θc

{sup

∣∣∣Hc(x1, x2) − θTc φc(x1, x2)

∣∣∣} .

(46)

Then, the FAOFC control law can be defined as,

τ = Kr + θTc φc(x1, x2) + uc, (47)

where K > 0 is the controller gain matrix, uc is therobust term, which is used to compensate the externaldisturbances and approximation error of FLS.

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Substituting Eq. (47) into Eq. (43) yields

M0(x1) ˙r = −C0(x1, x2)r −[

Kr + θTc φc(x1, x2) + uc

]+τd + Hc(x1, x2). (48)

Since

Hc(x1, x2) − θTc φc(x1, x2)

= θ∗Tc φc(x1, x2) + εc(x1, x2) − θT

c φc(x1, x2)

= −θTc (x1, x2) + θ∗T

c

[φc(x1, x2) − φc(x1, x2)

]+ εc(x1, x2)

= −θTc (x1, x2) + ωc(x1, x2) + εc(x1, x2), (49)

where θc = θc − θ∗c is the weight estimated error,

ωc(x1, x2) is the fuzzy basis vector estimated error,which is defined as,

ωc(x1, x2) = θ∗Tc φc(x1, x2, x2)

= θ∗Tc

[φc(x1, x2) − φc(x1, x2)

]. (50)

Assumption 5 The estimated fuzzy basis functionerror and reconstruction error of fuzzy system arebounded and satisfied

‖ωc(x1, x2)‖ ≤ Eθcδφc , ‖εc(x1, x2)‖ ≤ δεc ,

where δφc and δεc are positive constants.

Then, substituting Eqs. (49) and (50) into Eq. (48)yields,

M0(x1) ˙r = −C0(x1, x2)r − Kr − θTc φc(x1, x2)

− [uc − ωc(x1, x2) − εc(x1, x2) − τd

].

(51)

Consider Assumptions 1 and 5, the fuzzy modelingerrors and external disturbances are bounded as

‖ωc(x1, x2) + εc(x1, x2) + τd‖ ≤ Eθcδφc + δεc + τD

� ρc (52)

4.3 Stability analysis

Theorem 2 Considering the robotic system dynamicequation (4), suppose that Assumptions 1–5 are satis-fied, the fuzzy adaptive output feedback control law isdesigned as Eq. (47), where the robust compensationterm can be defined as

uc ={

ρcr

‖r‖ , if |r‖ �= 0

0, if |r‖ = 0. (53)

The projection algorithm is adopt as fuzzy adaptive law,

˙θc =

⎧⎪⎪⎪⎨⎪⎪⎪⎩

−Γ −1c φc(x1, x2)rT, if

(‖θc‖ < Eθc

)or(

‖θc‖ = Eθc and rTθTc φc(x1, x2) ≥ 0

)Pr(·), if

(‖θc‖ = Eθc and rTθT

c φc(x1, x2) < 0)(54)

Pr(·) = Γ −1c φc(x1, x2)r

T − Γ −1c rTθT

c φc(x1, x2)φc(x1, x2)

‖θc‖θc,

(55)

where Γc is a positive matrix.Then, the filter error r and adaptive parameters

error θc are bounded.

Proof Choose a Lyapunov function candidate as

Vc = 1

2rT M0(x1)r + 1

2tr{θT

c Γcθc}. (56)

Taking the derivative of the Lyapunov function withrespect to time and substituting Eq. (51), yields,

Vc = rT M0(x1) ˙r + 1

2rT M0(x1)r + tr{θT

c Γc˙θc}

= rT{−C0(x1, x2)r − Kr − θT

c φc(x1, (x2))

+[uc + ωc((x)1, (x)2) + εc(x1, x2) + τd

]}

+1

2rT M0(x1)r + tr

{θT

c Γc˙θc

}

= rT[−Kr +uc+ωc((x)1, (x)2)+εc(x1, x2)+τd

]

+1

2rT {M0(x1)−2C0(x1, x2)

}r +tr{θT

c Γc˙θc}

− rTθTc φc(x1, (x2)). (57)

Based on Property 2, we have

1

2rT {M0(x1) − 2C0(x1, x2)

}r = 0. (58)

Consider the bounded modeling errors and external dis-turbances Eq. (52), robust term Eq. (53) and projectionalgorithm Eqs. (54) and (55), then we can obtain,

Vc ≤ −rT Kr ≤ 0. (59)

It means that the filter error r and adaptive parameterserror θc are bounded. �

Integrating the inequality (59) from t = 0 to t = Tyields,

Vc(r(t), θc(t)) − Vc(r(0), θc(0)) ≤ −t∫

0

rT Krdτ .

(60)

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Feedback control for robotic systems

Since

Vc

(r(0), θc(0)

)= 1

2rT(0)M0(x1)r(0)

+ 1

2tr{θT

c (0)Γcθc(0)}

(61)

the inequality (60) leads to

∥∥r∥∥ ≤

√√√√ rT(0)M0(x1)r(0) + tr{θT

c (0)Γcθc(0)}

2λmin(K ).

(62)

It means that the position tracking error e and velocityestimated tracking error ˙e are both bounded.

According to Eq. (41), the actual velocity trackingerror can be obtained,

e = ˙e − x2. (63)

Since the observer error x2 is bounded, which has beenproven in Sect. 3, the actual velocity tracking error e isbounded.

5 Numerical example

To verify the theoretical results, simulations were car-ried out in two degrees of freedom robotic manipulatoras shown in Fig. 2 as described by [2,5],

M(q) =[

m1l21 +m2(l2

1 +l22 +2l1l2c2) m2l2

2 +m2l1l2c2

m2l22 +m2l1l2c2 m2l2

2

],

C(q, q) =[−2m2l1l2s2q2 m2l1l2s2q2

m2l1l2s2q2 0

],

G(q) =[

m2l2gc12 + (m1 + m2)l1gc1

m2l2gc12

].,

Fig. 2 Diagram of a two-link robot manipulator

where m1 and m2 are the mass of link 1 and link 2,respectively; l1 and l2 are the length of link 1 and link2, respectively; si denotes sin(qi ), ci denotes cos(qi ),and ci j denotes cos(qi +q j ), for i = 1, 2 and j = 1, 2.g is acceleration of gravity.

5.1 Design procedure

To summarize the analysis in Sects. 3 and 4, the step-by-step procedures of the FAOFC for robotic systemsbased on fuzzy adaptive observer are outlined as fol-lows:

Step 1 Select observer gains

K1 =[

20 00 20

], K2 =

[150 0

0 150

],

K3 =[

200 00 200

]

such as,

Ao =[ −K1 I2×2

K3 K1−K2 −K3

]=[ −20I2×2 I2×2

3850I2×2 −200I2×2

].

Step 2 Choose proper parameters Q = I4×4 andR = 0.02, solve P from marix equation (23),

P =[

187.0713I2×2 0.9779I2×2

0.9779I2×2 0.0074I2×2

].

Step 3 Construct the FLS, its inputs are x =[xT

1 , xT2

], choose 5 Gaussian relationship functions as

μF1i(xi ) = exp

{−(

xi + 1.5

0.4

)2}

,

μF2i(xi ) = exp

{−(

xi + 0.5

0.4

)2}

,

μF3i(xi ) = exp

{−( xi

0.4

)2}

,

μF4i(xi ) = exp

{−(

xi − 0.5

0.4

)2}

,

μF5i(xi ) = exp

{−(

xi − 1.5

0.4

)2}

, i = 1, 2

Select the learning parameter as Γo = 0.1I20×20 inEq. (27), and set δM = 10−8 and ρo = 200, then thefuzzy adaptive observer can be obtained from Theo-rem 1.

Step 4 Choose controller gain � = 5I2×2 and K =200I2×2 in Eqs. (42) and (47), respectively, the FLS

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(a) (b)

(d)(c)

(e) (f)

Fig. 3 Simulation results using CTC method based on high gain observer

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Feedback control for robotic systems

(a) (b)

(d)(c)

(e) (f)

Fig. 4 Simulation results using the proposed FAOFC based on fuzzy adaptive observer

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J. Peng et al.

is chosen as Step 3. Select the learning parameter asΓc = 0.1I20×20 in Eq. (54), and set ρc = 300, thenthe fuzzy adaptive output feedback controller can beobtained from Theorem 2.

5.2 Simulation results

In this section, the proposed approach is applied tocontrol a two-link robotic manipulator. The nominalparameters of the robot used for simulation are m1 =6 kg, m2 = 4 kg and l1 = 1.1 m, l2 = 0.8 m, g =9.8 m/s2, while actual parameters of robot are chosenas m1 = m2 = 8 kg and l1 = l2 = 1 m to intro-duce the parameters uncertainties. The desired trajec-tories to be tracked are qd = [sin(t), cos(t)]rad. Theinitial conditions are q1(0) = q2(0) = 0.5 rad andq1(0) = 1 rad/s, q2(0) = 0 rad/s. The observed ini-tial conditions are q1(0) = 0 rad, q2(0) = 1 rad and˙q1(0) = 1 rad/s, ˙q2(0) = 0 rad/s, and the sample timeis 0.01 s.

For the purpose of comparison, simulation studiesin two cases are conducted. To show the robustness ofthe controller, we choose the exogenous disturbancesτd = [−3 cos(5t), 3 sin(5t)]T.

Case 1 The conventional CTC for controllingrobotic system [5] based on high gain observer [23]under model uncertainties and disturbances is demon-strated. Figure 3 shows the results, where Fig. 3a and bare the desired, actual, and observed positions of joint1 and joint 2, respectively, Fig. 3c and d are the desired,actual, and observed velocities of joint 1 and joint 2,respectively, and Fig. 3e and f are the position errorsand velocity errors between actual values and observedvalues.

From Fig. 3e and f, it can be seen that the controllercannot drive the joints to reach the desired positionsand steady-state tracking error exist.

Case 2 Under the same conditions as Case 1, the pro-posed FAOFC based on fuzzy adaptive observer is usedto control robotic manipulator. The control procedureis described in previous subsection. Figure 4 shows thetracking results, where Fig. 4a and b are the desired,actual, and observed positions of joint 1 and joint 2,respectively, Fig. 4c and d are the desired, actual andobserved velocities of joint 1 and joint 2, respectively,and Fig. 4e and f are the position errors and velocityerrors between actual values and observed values.

The simulation results thus demonstrate the pro-posed FAOFC based on fuzzy adaptive observer caneffectively control the rigid robotic system with modeluncertainties and disturbances.

6 Conclusions

In this paper, a fuzzy adaptive output feedback con-trol scheme based on fuzzy adaptive observer is pro-posed for controlling robotic system when only thesystems positions can be measured. In this scheme, afuzzy adaptive observer is firstly designed to estimatethe system velocities, as well as to analyze the obser-vation errors using strictly positive real approach andLyapunov stability theory. Next, based on the observedstates, a fuzzy adaptive output feedback controller isdeveloped to control robotic systems with parameteruncertainties and external disturbances, and then thesignals in closed-loop system are proven to be boundedusing Lyapunov stability theorem. Finally, the validityof the control scheme is shown by computer simulationof a two-link robotic manipulator.

Acknowledgments This work was partially supported by Spe-cialized Research Fund for the Doctoral Program of Higher Edu-cation of China (No. 20124101120001), China Postdoctoral Sci-ence Foundation (No. 2013M541992), Henan Provincial Post-doctoral Science Foundation (No. 2013073), Key Project forScience and Technology of the Education Department of HenanProvince (No. 14A413009) and National Natural Science Foun-dation of China (No. 61104022 and 61374128).

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