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Research Papers

# Direct Adaptive Function Approximation Techniques Based Control of Robot ManipulatorsOPEN ACCESS

[+] Author and Article Information

Department of Electrical Engineering,
Behbahan Khatam Alanbia University
of Technology,
Behbahan 6361647189, Iran

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received July 18, 2016; final manuscript received June 10, 2017; published online September 5, 2017. Assoc. Editor: Azim Eskandarian.

J. Dyn. Sys., Meas., Control 140(1), 011006 (Sep 05, 2017) (11 pages) Paper No: DS-16-1351; doi: 10.1115/1.4037269 History: Received July 18, 2016; Revised June 10, 2017

## Abstract

In this paper, a simple model-free controller for electrically driven robot manipulators is presented using function approximation techniques (FAT) such as Legendre polynomials (LP) and Fourier series (FS). According to the orthogonal functions theorem, LP and FS can approximate nonlinear functions with an arbitrary small approximation error. From this point of view, they are similar to fuzzy systems and can be used as controller to approximate the ideal control law. In comparison with fuzzy systems and neural networks, LP and FS are simpler and less computational. Moreover, there are very few tuning parameters in LP and FS. Consequently, the proposed controller is less computational in comparison with fuzzy and neural controllers. The case study is an articulated robot manipulator driven by permanent magnet direct current (DC) motors. Simulation results verify the effectiveness of the proposed control approach and its superiority over neuro-fuzzy controllers.

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## Introduction

In the last few decades, model-free control of robot manipulators has been the focus of widespread researches. Reviewing the problems associated with model-based controllers reveals the importance and necessity of model-free controllers. The first problem is establishing and identifying a suitable mathematical model for the system. Suppose this stage is passed successfully. Then, the second problem arises which is sensing requirements. Model-based controller requires various feedbacks such as motor currents and/or joint accelerations. These signals are usually contaminated with noise and may degrade the controller performance. External disturbances, parametric uncertainty, and unmodeled dynamics are other challenges of model-based approaches.

Among model-free approaches, adaptive neuro-fuzzy control is very popular. During the last two decades, various neuro-fuzzy control approaches have been applied to robust control of many uncertain nonlinear systems [16]. The universal approximation property and the linear parameterization are the most important motivations for these widespread applications. Fault-tolerance, parallelism, and excellent learning capabilities are other beneficial characteristics of fuzzy systems and neural networks [7]. In neural control, the network structure, such as the number of layers and nodes, and the parameters in activation functions are important issues which should be determined carefully. In adaptive fuzzy control, there are many tuning parameters such as the center and width of the Gaussian membership functions and also the weight of each rule. Usually, these parameters are adjusted online using the adaptation laws derived from stability analysis. Nevertheless, the initial values of these parameters and their convergence rate are important issues that considerably affect the controller performance and should be selected carefully [8].

Recently, regressor-free control of robot manipulators has been presented which is based on function approximation techniques (FAT) [913]. In this approach, uncertainties such as the inertia matrix, Jacobian matrix, or gravity vector have been estimated using Fourier series (FS) expansion or Legendre polynomials (LP). In comparison with neuro-fuzzy control, this control strategy is simpler and less computational, since there are very fewer tuning parameters.

Most of presented works in the field of regressor-free control [913] are classified into indirect adaptive control. In indirect adaptive control, uncertainties in the ideal control law associated with the system dynamics are estimated, while in direct adaptive control where the ideal control law is estimated directly. Direct and indirect adaptive fuzzy controls have been extensively studied [1417].

Based on the orthogonal function theorem [18], LP and FS can approximate nonlinear functions. In other words, similar to neural networks and fuzzy systems, LP and FS are universal approximators. As a result, similar to direct adaptive fuzzy control in which fuzzy systems play the role of controller, LP and FS can be considered as controller in a direct adaptive scheme. Then, the Legendre coefficients or Fourier series coefficients can be tuned online using the adaptation laws derived from the stability analysis. In this study, this control idea is applied to position control of an electrically driven robot manipulator. As mentioned earlier, in comparison with neuro-fuzzy controllers, FAT-based controllers are simpler to tune. Moreover, in comparison with the previous FAT-based approaches [1012], the proposed controller in this paper is simpler, since there is no need to consider an approximator for each element of matrices introducing the manipulator dynamics such as inertia matrix. To show the superiority of the proposed method, a comparison has been performed using a neuro-fuzzy control.

This paper is organized into six sections. Following the Introduction, in Sec. 2, function approximation using LP and FS is described. Section 3 describes the mathematical model of the robotic system. Section 4 presents the proposed control law. The stability analysis is given in Sec. 5. Simulation results are presented in Sec. 6, and finally, Sec. 7 concludes the paper.

## Function Approximation Using Legendre Polynomials and Fourier Series

In a three-dimensional Cartesian coordinate, we can represent any arbitrary vector as a linear combination of unit vectors $i=[100]T$, $j=[010]T$, and $k=[001]T$, since they are linearly independent and also mutually orthogonal, due to the dot product which is generally an inner product. This idea can be simply extended to the space of nonlinear functions which results in an interesting technique for function approximation. Consider a typical inner product given by Display Formula

(1)$〈f,g〉=∫f*(x)g(x)dx$

in which $f*(x)$ is the complex conjugate of the function $f(x)$. If the inner product (1) is zero for $f(x)≠g(x)$, the functions $f(x)$ and $g(x)$ are called orthogonal. Suppose that $V$ is the space of all real-valued continuous-time functions. According to Kreyszig [18], a function $h(x)$ defined on the interval $[x1x2]$ in this space can be represented as Display Formula

(2)$h(x)=∑i=1maiφi(x)+εm(x)$

where the set ${φ1(x)…φm(x)}$ forms an orthogonal basis and $εm(x)$ is the approximation error. The coefficient $ai$ is calculated by Display Formula

(3)$ai=1Ai∫x1x2h(x)φi(x)dxi=0,1,…,m$
Display Formula
(4)$∫x1x2φi(x)φj(x)dx={0 i≠jAi i=j$

The approximation error $εm(x)$ is bounded in the sense that [19] Display Formula

(5)$limm→∞∫x1x2εm2(x)dx=0$

Considering the interval $[−1 1]$ and the inner product (1), the LP which are defined as Display Formula

(6)$φ0(x)=1$
Display Formula
(7)$φ1(x)=x$
Display Formula
(8)$(i+1)φi+1(x)=(2i+1)xφi(x)−iφi−1(x)i=1,…,m−1$

form an orthogonal basis [19]. Thus, a function $h(x)$ defined on the interval $[−1 1]$ can be approximated using LP in the form of Eq. (2) in which the coefficients $ai$, $i=0,1,…,m$, are calculated according to Eqs. (3) and (4), and the polynomials $φi(x)$$i=0,1,…,m$ are given by Eqs. (5)(8). Thus Display Formula

(9)$hLP(x)=∑i=1maiφi(x)=PTφ$

is the LP approximation of the function $h(x)$ in which Display Formula

(10)$P=[a0a1…am]T$
Display Formula
(11)$φ=[φ0φ1…φm]T$

In other words, $h(x)$ can be represented as Display Formula

(12)$h(x)=PTφ+εm$

Remark 1. The most important problem in applying orthogonal functions to control systems is that the function $h(x)$ is not available. Thus, the coefficients $ai$, $i=0,1,…,m$, cannot be calculated according to Eqs. (3) and (4), since $h(x)$ is unknown. In control systems, these coefficients are adjusted online using adaptation laws derived from the stability analysis. In Sec. 4, this issue will be explained in more detail.

Remark 2. Another important issue about using orthogonal functions for function approximation in control systems may be the fact that the functions $φi(x)$ are mutually orthogonal just on the interval $[−1 1]$. Out of this interval, the functions $φi(x)$ may not be mutually orthogonal. However, the uncertain functions which should be estimated in robust and adaptive control systems are generally functions of the variable time which may increase to infinity and cannot be limited to the interval $[−1 1]$. To solve this problem, we can let $x=sin(ωt)$ in which $ω$ is a predefined constant [19].

According to Khorashadizadeh and Fateh [20], if the function $F(t)$ defined on $[t1,t2]$ satisfies Dirichlet's conditions, then it can be expressed as Display Formula

(13)$F(t)=a0+∑k=1∞ak cos(ωkt)+bk sin(ωkt)$

where $a0$, $ak$, and $bk$ are the FS coefficients, $ωk=2kπ/T$ are the frequencies of sinusoidal functions, and $T$ is the fundamental period of $F(t)$. The truncation error is defined as Display Formula

(14)$εm(t)=F(t)−Fm(t)$

where $Fm(t)=a0+∑k=1mak cos(ωkt)+bk sin(ωkt)]$ is the Fourier series approximation. Note that $Fm(t)$ can be written as Display Formula

(15)$Fm(t)=PTφ(t)$

in which Display Formula

(16)$P=[a0a1b1…ambm]T$

(17)$φ(t)=[1 cos(ω1t) sin(ω1t)… cos(ωmt) sin(ωmt)]T$

## Mathematic Model of the Robotic System

The dynamic model of a robot manipulator can be described by $n$ generalized coordinates representing the degrees-of-freedoms of the $n$ joints as [21] by Display Formula

(18)$D(q)q¨+C(q,q˙)q˙+G(q)=τl$

where $q∈Rn$ is the vector of joint positions, $D(q)$ the $n×n$ matrix of manipulator inertia, $C(q,q˙)q˙∈Rn$ the vector of centrifugal and Coriolis torques, $g(q)∈Rn$ the vector of gravitational torques, and $τl∈Rn$ the joint torque vector of robot. The electric motors provide the joint torque vector with the following function: Display Formula

(19)$Jq¨+Bq˙+rτl=τm$

where $τm∈Rn$ is the torque vector of motors; $J$, $B$, and $r$ are the $n×n$ diagonal matrices for motor coefficients namely the inertia, damping, and reduction gear, respectively. The joint velocity vector $q˙$ and the motor velocity vector $q˙m∈Rn$ are related through the gears to yield Display Formula

(20)$rq˙m=q˙$

In order to obtain the motor voltages as the inputs of system, we consider the electrical equation of geared permanent magnet direct current (DC) motors in the matrix form Display Formula

(21)$RIa+LI˙a+Kbr−1q˙=v$

where $v∈Rn$ is the vector of motor voltages and $Ia∈Rn$ is the vector of motor currents. The matrices $R$, $L$, and $Kb$ represent the $n×n$ diagonal matrices for the coefficients of armature resistance, inductance, and back-emf constant, respectively. The motor torque vector $τm$ as the input for dynamic (19) is produced by the motor current vector Display Formula

(22)$KmIa=τm$

where $Km$ is a diagonal matrix of the torque constants. In order to apply the independent joint technique, consider the voltage Eq. (21) in the scalar form as [22] Display Formula

(23)$RIa+LI˙a+kbr−1q˙=v$

Considering the term $LI˙a$, the system will be a third-order system. As a result, the required feedbacks will increase. To solve this problem, we can ignore the term $LI˙a$ and rewrite Eq. (23) as Display Formula

(24)$RIa+kbr−1q˙+ϕ(t)=v$

in which $ϕ(t)=LI˙a$. Using Eqs. (19) and (22) in scalar forms, it follows that Display Formula

(25)$Ia=km−1(Jr−1q¨+Br−1q˙+rτl)$

Substituting Eq. (25) into Eq. (24) yields Display Formula

(26)$Rkm−1(Jr−1q¨+Br−1q˙+rτl)+kbr−1q˙+ϕ(t)=v$

which can be rewritten as Display Formula

(27)$C1q¨+C2q˙+η(t)=v$

where $C1=Rkm−1Jr−1$, $C2=Rkm−1Br−1+kbr−1$, and $η(t)=Rkm−1rτl+ϕ(t)$. Since the actual values of $C1$ and $C2$ are unknown, we can consider the following nominal model: Display Formula

(28)$Ĉ1q¨+Ĉ2q˙+η1(t)=v$

in which $η1(t)=η(t)+(C1−Ĉ1)q¨+(C2−Ĉ2)q˙$. We can design the controller based on the nominal model (28). However, it requires the nominal values of $R̂$, $k̂m$, $Ĵ$, $r̂$, $B̂$, and $k̂b$. In this paper, we assume that these nominal values are not available. Thus, in order to eliminate the need for nominal model, we can consider Eq. (28) as [22] Display Formula

(29)$v=q¨−f(t)$

where $f(t)=(Ĉ1−1)q¨+Ĉ2q˙+η1(t)$.

## The Proposed Control Law

Define the tracking error as $e=qd−q$. Using feedback linearization technique, the ideal control law is given by Display Formula

(30)$v∗=q¨d+kTE−f(t)$

in which $E=[e e˙]T$ and $k=[k1 k2]T$ are gain vectors defined by the designer. Since $f(t)$ is uncertain, this control law cannot be implemented. However, similar to direct adaptive fuzzy control [23], we can assume that this ideal control law can be approximated by a LP or a FS as Display Formula

(31)$v*=PTφ(t)+εm(t)$

where $PTφ(t)$ is the best LP or FS that approximate $v*$ with the smallest approximation error denoted by $εm(t)$. It should be emphasized that since $v*$ is not available, the vector $P$ is unknown. Therefore, the control law is estimated as Display Formula

(32)$v=P̂Tφ(t)+vc(t)$

in which $P̂$ is the estimation of $P$ and the term $vc(t)$ compensates for the truncation error $εm(t)$. Substitution of the control law (32) into the system dynamics (29) yields the following closed-loop system: Display Formula

(33)$q¨=P̂Tφ(t)+vc(t)+f(t)$

According to Eq. (30), $f(t)$ can be given by Display Formula

(34)$f(t)=q¨d+kTE−v∗$

Using Eqs. (33) and (34), the closed-loop system can be rewritten as Display Formula

(35)$q¨=P̂Tφ(t)+vc(t)+q¨d+kTE−v∗$

In other words Display Formula

(36)$e¨=−kTE+P̃Tφ(t)−vc+εm$

where $P̃=P−P̂$. The closed-loop system (36) can also be represented by the following state-space equations: Display Formula

(37)$E˙=AE+Bw$
Display Formula
(38)$A=[01−k1−k2], B=[01], w=P̃Tφ(t)−vc+εm$

## Stability Analysis

A proof for the boundedness of internal signals is given by stability analysis and the adaptation laws for the FS coefficients or LP are designed. In order to analyze the stability, the following assumptions are made.

Assumption 1. The desired trajectory is smooth in the sense that$qd$ and its derivatives up to a necessary order are available and all uniformly bounded [21].

Assumption 2. The truncation error vector is bounded [19] as$|εm|≤ρ$ where$ρ$ is a known constant.

Assumption 3. There exist positive definite matrices$P0$ and$Q$ such that [24]Display Formula

(39)$ATP0+P0A=−Q$

where $A$ is a Hurwitz matrix given in Eq. (38).

Assumption 4.Motors are protected against high voltages as$|v|≤vmax$ [3].

Theorem 1.Consider the dynamical system (29) and the control law (32). The internal signals are bounded and the error vector$E$ asymptotically converges to zero if the following adaptation law for the FS coefficients or Legendre coefficients holds:Display Formula

(40)$P̂˙=γφ(t)ETP0B$

In Eq. (40), the positive constant of $γ$ determines the convergence rate and $P0$ is the positive definite matrix explained in Assumption 3.

Proof. Consider the following Lyapunov function candidate: Display Formula

(41)$V(t)=12ETP0E+12γP̃TP̃$
in which $P0$ is a positive definite matrix. The time derivative of Eq. (41) is given by Display Formula
(42)$V˙(t)=12E˙TP0E+12ETP0E˙+1γP̃˙TP̃$

Substitution of $E˙$ from Eq. (37) into Eq. (42) and using Assumption 3 yields Display Formula

(43)$V˙(t)=−12ETQE+ETP0Bw−1γP̃TP̂˙$

Substitution of $P̂˙$ from Eq. (40) and $w$ from Eq. (38) into Eq. (43) obtains Display Formula

(44)$V˙(t)=−12ETQE+ETP0B(−vc+εm)$

Now, in order to make $V˙(t)≤0$, the robust control term $vc$ should be determined so that the inequality Display Formula

(45)$ETP0B(−vc+εm)≤0$

is satisfied. Suppose that $vc$ is defined as Display Formula

(46)$vc=ρsign(ETP0B)$

As a result, Eq. (45) can be rewritten as Display Formula

(47)$−ρETP0Bsign(ETP0B)+ETP0Bεm≤0$

In other words, Display Formula

(48)$|ETP0B|(|εm|−ρ)≤0$

Using Assumption 2, it is simply verified that the inequality (48) holds. Therefore, $V˙(t)≤0$ is satisfied. Consequently, we have [24] Display Formula

(49)$V(P̃(t),E(t))≤V(P̃(0),E(0))$

which implies that $P̃(t)$ and $E(t)$ are bounded. Let us define $Ω(t)=−(1/2)ETQE$ [20]. It is obvious that $Ω(t)≤−V˙$. Integrating it with respect to time yields Display Formula

(50)$∫0tΩ(τ)dτ≤V(P̃(0),E(0))−V(P̃(t),E(t))$

Because $V(P̃(0),E(0))$ is bounded and $V(P̃(t),E(t))$ is nonincreasing and bounded, the following result is obtained: Display Formula

(51)$limt→∞∫0tΩ(τ)dτ≤∞$

According to Eq. (37), it is concluded that $E˙$ is bounded. Thus, $Ω˙(t)=−E˙TQE$ is also bounded. Now, using Barbalat's lemma, it can be shown that $limt→∞ Ω(t)=0$. Therefore, the asymptotic convergence of the tracking error and also its time derivative is proven. Due to Assumption 4, motor voltage is bounded. According to a proof given in Ref. [25], in an electrically driven robot, when the motor voltage is bounded, the motor currents $Ia$ and $I˙a$ are bounded. Thus, the internal signals are bounded.▪

To attenuate the chattering problem caused by the sign function, the following algorithm is proposed [26]. When the scalar $ETP0B$ is within the boundary $|ETP0B|≤E0$ ($E0$ is the thickness of the boundary layer), a PI structure Display Formula

(52)$vc(t)=[kp(t)ki(t)][ETP0B∫0τET(τ)P0Bdτ]=αTψ$

in which $αT=[kp(t)ki(t)]$ and $ψ=[ETP0B∫0τET(τ)P0Bdτ]T$ are applied and when $ETP0B$ is outside the boundary layer, $|ETP0B|≤E0$, the controller output is kept at the saturated value of $ρ$, i.e., $vc(t)=ρsign(ETP0B)$. It is assumed that the gain vector $α$ is not known in advance and will be tuned online. As a result, an adaptation law is required for $α̂$ which is designed using the following theorem.

Theorem 2.Consider the dynamical system (29) and the control law (32). The internal signals are bounded and the error vector$E$ asymptotically converges to zero if the following adaptation laws hold:Display Formula

(53)$P̂˙=γφ(t)ETP0B$
Display Formula
(54)$α̂˙=γ1ψETP0B$

Proof. Consider the Lyapunov function candidate as Display Formula

(55)$V(t)=12ETP0E+P̃TP̃2γ+α̃Tα̃2γ1$

where $α̃=α∗−α̂$ and $α∗$ is the optimal value for $α$ defined as Display Formula

(56)$α*=arg minα̂∈ℜ2 (supETP0B∈R |α̂Tψ−ρsign(ETP0B)|)$

The time derivative of Eq. (55) is given by Display Formula

(57)$V˙=12E˙TP0E+12ETP0E˙−P̃TP̂˙γ−α̃Tα̂˙γ1$

Substitution of $E˙$ from Eq. (37) into Eq. (58) and using Assumption 3 yields Display Formula

(58)$V˙=−12ETQE+ETP0Bw−P̃TP̂˙γ−α̃Tα̂˙γ1$

For simplicity, let us define $s=ETP0B$. Substitution of $P̂˙$ from Eq. (53) and $w$ from Eq. (38) into Eq. (58) obtains Display Formula

(59)$V˙=−12ETQE+s(−α̂Tψ+εm)−α̃Tα̂˙γ1$

Adding and subtracting $sα*Tψ$ to the right-hand side of Eq. (59) results in Display Formula

(60)$V˙=−ETQE2+s(−α̂Tψ+εm+α*Tψ−α*Tψ)−α̃Tα̂˙γ1$

which can be simplified as Display Formula

(61)$V˙=−ETQE2+s(α̃Tψ+εm−ρsign(s))−α̃Tα̂˙γ1$

Substitution of $α̂˙$ from Eq. (54) into Eq. (61) yields Display Formula

(62)$V˙=−ETQE2+s(εm−ρsign(s))$
Now, in order to make $V˙(t)≤0$, we should verify that the following inequality holds: Display Formula
(63)$s(εm−ρsign(s))≤0$

Thus, we can write Display Formula

(64)$sεm−ρ|s|≤|s|(|εm|−ρ)≤0$

Using Assumption 2, it is simply verified that the inequality (64) holds. Therefore, $V˙(t)≤0$ is satisfied. Then, similar to the reasoning presented for the previous theorem, it is concluded that $E$ asymptotically converges to zero and internal signals are bounded.

## Simulation Results

The proposed control law in Eq. (32) is simulated using an articulated robot driven by permanent magnet dc motors. The robot dynamical and kinematical parameters and also motor parameters have been completely described in Ref. [3]. The maximum voltage of each motor is set to $vmax=40 V$. The desired trajectory should be sufficiently smooth such that all its derivatives up to the required order are bounded. The desired position for every joint is defined as Display Formula

(65)$qdi=1−cos(πt/10) i=1,2,3$

###### Simulation 1: Uncertainty Estimation Using Legendre Polynomials.

In this simulation, the performance of direct FAT-based control using LP is studied. Suppose that the first five terms of LP defined by Display Formula

(66)$φ=[1x12(3x2−1)12(5x3−3x)18(35x4−30x2+3)]T , x=sin(0.2t)$

have been used as controller in each joint. The initial values of all Legendre coefficients $P̂(0)$ are set to zero. The convergence rates $γ$ and $γ1$ are set to 300 and 1, respectively. The reason of these choices is obvious. Since Legendre coefficients are far from their unknown suitable values, the convergence rates $γ$ should be large to make the changes applied to these coefficients larger. Also, since it is assumed that the truncation error is small, the convergence rate $γ1$ has been selected small. The matrices $A$ in Eq. (38) and $Q$ in Eq. (39) have been selected as Display Formula

(67)$A=[01−1−2], Q=[40000400]$

Then, the matrix $P0$ in Eq. (39) is calculated using the following matlab command: Display Formula

(68)$P0=lyap(AT,Q)=[600200200200]$

In order to have better comparisons, consider the following cost function: Display Formula

(69)$J=∫040‖e(t)‖dt$

Using these settings, the controller has been applied to the robotic system. The tracking error is illustrated in Fig. 1. As shown in this figure, the tracking errors start from a non-zero value and converge to zero without any overshoot. The tracking performance is plotted in Fig. 2. According to this figure, after 5 s, joint variables converge to the desired trajectory and maintain on it. Motor voltages as control efforts are presented in Fig. 3. As it can be seen, motor voltages are smooth and without any chattering. Also, they are bounded and within the permitted range. In this case, the cost function takes the value of $J=0.9423$. To study the influence of increasing the number of LP on the cost function, 16 terms have been used. The result is $J=0.9384$. Thus, increasing the number of LP cannot improve the controller performance considerably. The reason is that the truncation error has been compensated and there is no need to increase the size of vector $φ(t)$. To examine the controller performance in rejecting the effects of external disturbances, the following signal has been applied to all joints of the robotic system:

Display Formula

(70)$vdist(t)=8u(t−8)−9u(t−16)+10u(t−27)$

This signal affects the system in the form of Display Formula

(71)$RIa+LI˙a+kbr−1q˙+vdist(t)=v$

The tracking errors in the presence of this external disturbance are illustrated in Fig. 4. In comparison with the control effort presented in Fig. 3, it is obvious that the external disturbance amplitude is very large. In fact, it is 66% of the control signal. Nevertheless, as shown in Fig. 4, this large external disturbance has been rejected effectively and its influence on the robotic system is negligible. In this case, the cost function takes the value of $J=0.9424$ using five LP.

###### Simulation 2: Uncertainty Estimation Using the Fourier Series Expansion.

In this simulation, the performance of direct regressor-free control using the FS expansion is studied. Suppose that the first five terms of FS defined by Display Formula

(72)$φ(t)=[1 cos(ω1t) sin(ω1t) cos(2ω1t) sin(2ω1t)]T$

have been used as controller in each joint. All of the controller parameter values are the same as those described in simulation 1. The parameter $ω1$ has been set to $π/10$. Figure 5 shows the tracking error. According to this figure, the tracking errors are bounded and converge to zero without any overshoot. The tracking performance is plotted in Fig. 6. According to this figure, after 5 s, joint variables converge to the desired trajectory and maintain on it. Motor voltages as control efforts are presented in Fig. 7. As it can be seen, motor voltages are smooth and without any chattering. Also, they are bounded and within the permitted range. The results obtained by FS (Figs. 57) and LP (Figs. 13) are nearly the same and the differences are negligible. The reason is that both LP and FS have been tuned well and are estimating the ideal control law satisfactorily. In this case, the cost function takes the value of $J=0.9347$. To study the influence of increasing the size of the vector $φ(t)$ on the cost function, nine terms have been used. In other words, the vector $φ(t)$ is changed to

Display Formula

(73)$φ(t)=[1 cos(ω1t) sin(ω1t) cos(2ω1t) sin(2ω1t) cos(3ω1t) sin(3ω1t) cos(4ω1t) sin(4ω1t)]T$

The result is $J=0.9331$. Thus, increasing the number of sinusoidal terms in the vector $φ(t)$ cannot improve the controller performance considerably. As mentioned previously, the reason may be compensation of the truncation error in the control law. Nevertheless, increasing the size of this vector in both cases (LP and FS) reduces the tracking error. To examine the controller performance in rejecting the effects of external disturbances, the signal given by Eq. (70) has been applied to all joints of the robotic system. The tracking errors in the presence of this external disturbance are illustrated in Fig. 8. In this case, the cost function takes the value of $J=0.9377$ using the vector $φ(t)$ given by (72)

Display Formula

(74)$RIa+LI˙a+kbr−1q˙+vdist(t)=v$

###### Simulation 3: The Performance of a Neuro-Fuzzy Controller.

Consider the neuro-fuzzy controller described in Ref. [27]. The block diagram of this controller is also given in Ref. [27]. As explained in Ref. [27], the first layer is the input layer with the input variable of $ei, 1≤i≤n$. The error function is in the form of Display Formula

(75)$e=Kp(qd−q)+Kd(q˙d−q˙)+(q¨d−q¨)$

The parameters $Kp$ and $Kd$ are positive definite matrices. The second layer is the membership layer, the third layer is the rule layer, the fourth layer is the Takagi-Sugeno-Kang layer, and the last layer is the output layer as described in the mentioned reference.

This fuzzy-neural network controller is applied to the robotic system described in this paper. The parameters $Kp$ and $Kd$ have been set to 25 and 10, respectively. The tracking errors of this controller are presented in Fig. 9. A comparison between Figs. 1 and 5 reveals the superiority of the proposed method. Although increasing the controller gains $Kp$ and $Kd$ reduces the tracking errors, it degrades the quality of the control signal by the chattering phenomenon. Motor voltages using the fuzzy-neural network have been presented in Fig. 10. The chattering phenomenon is completely clear in this figure, while the control signal using LP (Fig. 3) and FS (Fig. 7) is without chattering. If we reduce the gains $Kp$ and $Kd$, the chattering phenomenon will be avoided. However, the tracking errors will not be satisfactory. As can be seen, this controller is too computational, while the proposed method is simpler and less computational.

## Conclusion

Recently, some FAT-based control approaches have been presented in literatures for robust tracking control of robot manipulators. These approaches are based on uncertainty estimation using FS and LP, since they are universal approximators. It should be emphasized that the previous related works using FS and LP are classified in the category of indirect adaptive control. The novelty of this paper is presenting a direct adaptive FAT-based control in which the ideal control law is directly estimated using FS or LP. Simulation results verify the effectiveness of the proposed controller. Also, a comparison between the proposed controller and a neuro-fuzzy controller has been performed that shows the simplicity and efficiency of the proposed method.

## Acknowledgements

The author gratefully appreciates the support of the Behbahan Khatam Alanbia University of Technology.

## References

Li, Y. , Tong, S. , and Li, T. , 2012, “ Fuzzy Adaptive Dynamic Surface Control for a Single-Link Flexible-Joint Robot,” Nonlinear Dyn., 70(3), pp. 2035–2048.
Moradi Zirkohi, M. , Fateh, M. M. , and Shoorehdeli, M. A. , 2013, “ Type-2 Fuzzy Control for a Flexible-Joint Robot Using Voltage Control Strategy,” Int. J. Autom. Comput., 10(3), pp. 242–255.
Fateh, M. M. , and Khorashadizadeh, S. , 2012, “ Robust Control of Electrically Driven Robots by Adaptive Fuzzy Estimation of Uncertainty,” Nonlinear Dyn., 69(3), pp. 1465–1477.
Puga-Guzmán, S. , Moreno-Valenzuela, J. , and Santibáñez, V. , 2014, “ Adaptive Neural Network Motion Control of Manipulators With Experimental Evaluations,” Sci. World J., 2014, p. 694706.
Zhai, D. H. , and Xia, Y. , 2016, “ Adaptive Fuzzy Control of Multilateral Asymmetric Teleoperation for Coordinated Multiple Mobile Manipulators,” IEEE Trans. Fuzzy Syst., 24(1), pp. 57–70.
Tong, S. , Shuai, S. , and Yongming, Li. , 2015, “ Fuzzy Adaptive Output Feedback Control of MIMO Nonlinear Systems With Partial Tracking Errors Constrained,” IEEE Trans. Fuzzy Syst., 23(4), pp. 729–742.
Khorashadizadeh, S. , and Fateh, M. M. , 2015, “ Uncertainty Estimation in Robust Tracking Control of Robot Manipulators Using the Fourier Series Expansion,” Robotica, 35(2), pp. 310–336.
Khorashadizadeh, S. , and Mahdian, M. , 2016, “ Voltage Tracking Control of DC-DC Boost Converter Using Brain Emotional Learning,” Fourth International Conference on Control, Instrumentation, and Automation (ICCIA), Qazvin, Iran, Jan. 27–28, pp. 268–272.
Tsai, Y. C. , and Huang, A. C. , 2008, “ FAT-Based Adaptive Control for Pneumatic Servo Systems With Mismatched Uncertainties,” Mech. Syst. Signal Process., 22(6), pp. 1263–1273.
Huang, A. C. , Wu, S. C. , and Ting, W. F. , 2006, “ A FAT-Based Adaptive Controller for Robot Manipulators Without Regressor Matrix: Theory and Experiments,” Robotica, 24(2), pp. 205–210.
Chien, M. C. , and Huang, A. C. , 2012, “ Adaptive Impedance Controller Design for Flexible-Joint Electrically-Driven Robots Without Computation of the Regressor Matrix,” Robotica, 30(1), pp. 133–144.
Chien, M. C. , and Huang, A. C. , 2010, “ Design of a FAT-Based Adaptive Visual Servoing for Robots With Time Varying Uncertainties,” Int. J. Optomechatronics, 4(2), pp. 93–114.
Fard, M. B. , and Khorashadizadeh, S. , 2015, “ Model Free Robust Impedance Control of Robot Manipulators Using Fourier Series Expansion,” AI & Robotics (IRANOPEN), Qazvin, Iran. Apr. 12, pp. 1–7.
Wang, C. H. , Liu, H. L. , and Lin, T. C. , 2002, “ Direct Adaptive Fuzzy-Neural Control With State Observer and Supervisory Controller for Unknown Nonlinear Dynamical Systems,” IEEE Trans. Fuzzy Syst., 10(1), pp. 39–49.
Hsueh, Y. C. , and Su, S. F. , 2012, “ Learning Error Feedback Design of Direct Adaptive Fuzzy Control Systems,” IEEE Trans. Fuzzy Syst., 20(3), pp. 536–545.
Li, Y. , Liand, T. , and Jing, X. , 2014, “ Indirect Adaptive Fuzzy Control for Input and Output Constrained Nonlinear Systems Using a Barrier Lyapunov Function,” Int. J. Adapt. Control Signal Process., 28(2), pp. 184–199.
Boulkroune, A. , Bounar, N. , and Farza, M. , 2014, “ Indirect Adaptive Fuzzy Control Scheme Based on Observer for Nonlinear Systems: A Novel SPR-Filter Approach,” Neurocomputing, 135, pp. 378–387.
Kreyszig, E. , 2007, Advanced Engineering Mathematics, Wiley, New York. [PubMed] [PubMed]
Khorashadizadeh, S. , and Fateh, M. M. , 2015, “ Robust Task-Space Control of Robot Manipulators Using Legendre Polynomials for Uncertainty Estimation,” Nonlinear Dyn., 79(2), pp. 1151–1161.
Khorashadizadeh, S. , and Fateh, M. M. , 2013, “ Adaptive Fourier Series-Based Control of Electrically Driven Robot Manipulators,” Third International Conference on Control, Instrumentation, and Automation (ICCIA), Tehran, Iran, Dec. 28–30, pp. 213–218.
Spong, M. W. , Hutchinson, S. , and Vidyasagar, M. , 2006, Robot Modelling and Control, Wiley, Hoboken, NJ.
Fateh, M. M. , Azargoshasb, S. , and Khorashadizadeh, S. , 2014, “ Model-Free Discrete Control for Robot Manipulators Using a Fuzzy Estimator,” COMPEL: Int. J. Comput. Math. Electr. Electron. Eng., 33(3), pp. 1051–1067.
Wang, L. X. , 1994, Adaptive Fuzzy Systems and Control: Design and Stability Analysis, Prentice Hall, Upper Saddle River, NJ.
Slotine, J. J. , and Li, W. , 1991, Applied Nonlinear Control, Vol. 199, Prentice Hall, Englewood Cliffs, NJ.
Fateh, M. M. , 2012, “ Robust Control of Flexible-Joint Robots Using Voltage Control Strategy,” Nonlinear Dyn., 67(2), pp. 1525–1537.
Shahnazi, R. , Modir Shanechi, H. , and Pariz, N. , 2008, “ Position Control of Induction and DC Servomotors: A Novel Adaptive Fuzzy PI Sliding Mode Control,” IEEE Trans. Energy Convers., 23(1), pp. 138–147.
Wai, R. J. , and Chen, P. C. , 2004, “ Intelligent Tracking Control for Robot Manipulator Including Actuator Dynamics Via TSK-Type Fuzzy Neural Network,” IEEE Trans. Fuzzy Syst., 12(4), pp. 552–559.
View article in PDF format.

## References

Li, Y. , Tong, S. , and Li, T. , 2012, “ Fuzzy Adaptive Dynamic Surface Control for a Single-Link Flexible-Joint Robot,” Nonlinear Dyn., 70(3), pp. 2035–2048.
Moradi Zirkohi, M. , Fateh, M. M. , and Shoorehdeli, M. A. , 2013, “ Type-2 Fuzzy Control for a Flexible-Joint Robot Using Voltage Control Strategy,” Int. J. Autom. Comput., 10(3), pp. 242–255.
Fateh, M. M. , and Khorashadizadeh, S. , 2012, “ Robust Control of Electrically Driven Robots by Adaptive Fuzzy Estimation of Uncertainty,” Nonlinear Dyn., 69(3), pp. 1465–1477.
Puga-Guzmán, S. , Moreno-Valenzuela, J. , and Santibáñez, V. , 2014, “ Adaptive Neural Network Motion Control of Manipulators With Experimental Evaluations,” Sci. World J., 2014, p. 694706.
Zhai, D. H. , and Xia, Y. , 2016, “ Adaptive Fuzzy Control of Multilateral Asymmetric Teleoperation for Coordinated Multiple Mobile Manipulators,” IEEE Trans. Fuzzy Syst., 24(1), pp. 57–70.
Tong, S. , Shuai, S. , and Yongming, Li. , 2015, “ Fuzzy Adaptive Output Feedback Control of MIMO Nonlinear Systems With Partial Tracking Errors Constrained,” IEEE Trans. Fuzzy Syst., 23(4), pp. 729–742.
Khorashadizadeh, S. , and Fateh, M. M. , 2015, “ Uncertainty Estimation in Robust Tracking Control of Robot Manipulators Using the Fourier Series Expansion,” Robotica, 35(2), pp. 310–336.
Khorashadizadeh, S. , and Mahdian, M. , 2016, “ Voltage Tracking Control of DC-DC Boost Converter Using Brain Emotional Learning,” Fourth International Conference on Control, Instrumentation, and Automation (ICCIA), Qazvin, Iran, Jan. 27–28, pp. 268–272.
Tsai, Y. C. , and Huang, A. C. , 2008, “ FAT-Based Adaptive Control for Pneumatic Servo Systems With Mismatched Uncertainties,” Mech. Syst. Signal Process., 22(6), pp. 1263–1273.
Huang, A. C. , Wu, S. C. , and Ting, W. F. , 2006, “ A FAT-Based Adaptive Controller for Robot Manipulators Without Regressor Matrix: Theory and Experiments,” Robotica, 24(2), pp. 205–210.
Chien, M. C. , and Huang, A. C. , 2012, “ Adaptive Impedance Controller Design for Flexible-Joint Electrically-Driven Robots Without Computation of the Regressor Matrix,” Robotica, 30(1), pp. 133–144.
Chien, M. C. , and Huang, A. C. , 2010, “ Design of a FAT-Based Adaptive Visual Servoing for Robots With Time Varying Uncertainties,” Int. J. Optomechatronics, 4(2), pp. 93–114.
Fard, M. B. , and Khorashadizadeh, S. , 2015, “ Model Free Robust Impedance Control of Robot Manipulators Using Fourier Series Expansion,” AI & Robotics (IRANOPEN), Qazvin, Iran. Apr. 12, pp. 1–7.
Wang, C. H. , Liu, H. L. , and Lin, T. C. , 2002, “ Direct Adaptive Fuzzy-Neural Control With State Observer and Supervisory Controller for Unknown Nonlinear Dynamical Systems,” IEEE Trans. Fuzzy Syst., 10(1), pp. 39–49.
Hsueh, Y. C. , and Su, S. F. , 2012, “ Learning Error Feedback Design of Direct Adaptive Fuzzy Control Systems,” IEEE Trans. Fuzzy Syst., 20(3), pp. 536–545.
Li, Y. , Liand, T. , and Jing, X. , 2014, “ Indirect Adaptive Fuzzy Control for Input and Output Constrained Nonlinear Systems Using a Barrier Lyapunov Function,” Int. J. Adapt. Control Signal Process., 28(2), pp. 184–199.
Boulkroune, A. , Bounar, N. , and Farza, M. , 2014, “ Indirect Adaptive Fuzzy Control Scheme Based on Observer for Nonlinear Systems: A Novel SPR-Filter Approach,” Neurocomputing, 135, pp. 378–387.
Kreyszig, E. , 2007, Advanced Engineering Mathematics, Wiley, New York. [PubMed] [PubMed]
Khorashadizadeh, S. , and Fateh, M. M. , 2015, “ Robust Task-Space Control of Robot Manipulators Using Legendre Polynomials for Uncertainty Estimation,” Nonlinear Dyn., 79(2), pp. 1151–1161.
Khorashadizadeh, S. , and Fateh, M. M. , 2013, “ Adaptive Fourier Series-Based Control of Electrically Driven Robot Manipulators,” Third International Conference on Control, Instrumentation, and Automation (ICCIA), Tehran, Iran, Dec. 28–30, pp. 213–218.
Spong, M. W. , Hutchinson, S. , and Vidyasagar, M. , 2006, Robot Modelling and Control, Wiley, Hoboken, NJ.
Fateh, M. M. , Azargoshasb, S. , and Khorashadizadeh, S. , 2014, “ Model-Free Discrete Control for Robot Manipulators Using a Fuzzy Estimator,” COMPEL: Int. J. Comput. Math. Electr. Electron. Eng., 33(3), pp. 1051–1067.
Wang, L. X. , 1994, Adaptive Fuzzy Systems and Control: Design and Stability Analysis, Prentice Hall, Upper Saddle River, NJ.
Slotine, J. J. , and Li, W. , 1991, Applied Nonlinear Control, Vol. 199, Prentice Hall, Englewood Cliffs, NJ.
Fateh, M. M. , 2012, “ Robust Control of Flexible-Joint Robots Using Voltage Control Strategy,” Nonlinear Dyn., 67(2), pp. 1525–1537.
Shahnazi, R. , Modir Shanechi, H. , and Pariz, N. , 2008, “ Position Control of Induction and DC Servomotors: A Novel Adaptive Fuzzy PI Sliding Mode Control,” IEEE Trans. Energy Convers., 23(1), pp. 138–147.
Wai, R. J. , and Chen, P. C. , 2004, “ Intelligent Tracking Control for Robot Manipulator Including Actuator Dynamics Via TSK-Type Fuzzy Neural Network,” IEEE Trans. Fuzzy Syst., 12(4), pp. 552–559.

## Figures

Fig. 1

The tracking errors using LP

Fig. 2

The tracking performance using LP

Fig. 3

Motor voltages using LP

Fig. 4

The tracking errors using LP in the presence of external disturbance

Fig. 5

The tracking errors using FS

Fig. 6

The tracking performance using FS

Fig. 7

Motor voltages using FS

Fig. 8

The tracking errors using FS in the presence of external disturbance

Fig. 9

The tracking errors of fuzzy-neural network

Fig. 10

The control signals of fuzzy-neural network

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