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FUZZY LOGIC-BASED ADAPTIVE TRACKING CONTROL OF MANIPULATOR ACTUATED BY DC MOTOR
Jinglei Zhou, Zhenwu Li, Guizhi Lv, and Endong Liu
References
[1] K.J. Cheng and P.Y. Cheng, Investigation on velocity perfor-mance deviation of serial manipulators resulted from fabrica-tion errors, International Journal of Robotics and Automation,32(4), 2017, 324–332.
[2] Z.X. Yang and D. Zhang, Energy optimal adaption and mo-tion planning of a 3-RRS balanced manipulator, Interna-tional Journal of Robotics and Automation, 34(5), 2019, doi:10.2316/J.2019.206-0171.
[3] D. Cafolla, M. Wang, G. Carbone, et al., LARMbot: A newhumanoid robot with parallel mechanisms, Proc. 21st CISM-IFToMM Symposium, Udine, Italy, 2016, 275–283.
[4] M. Russo, D. Cafolla and M. Ceccarelli, Development ofLARMbot 2, a novel humanoid robot with parallel architec-tures, Proc. 4th IFToMM Symposium on Mechanism Designfor Robotics, Udine, Italy, 2018, 17–24.
[5] J. Chen and H. Lau, Policy gradient-based inverse kinematicsreﬁnement for tendon-driven serpentine surgical manipulator,International Journal of Robotics and Automation, 34(3), 2019,doi: 10.2316/J.2019.206-5461.
[6] F. Yang, G.L. Zhang, L. Yuan, et al., End-eﬀector optimaltracking control of free-ﬂoating space robot, Journal of Astro-nautics, 37(7), 2016, 846–853.
[7] J.Y. Li, L. Wan, H. Huang, et al., Adaptive control methodof underwater vehicle manipulator system under disturbances,Journal of Tianjin University (Science and Technology), 51(4),2018, 413–421.
[8] C.J. Huang, U. Farooq, H.Y. Liu, et al., A PSO-tuned fuzzylogic system for position tracking of mobile robot, Interna-tional Journal of Robotics and Automation, 34(1), 2019, doi:10.2316/J.2019.206-5422.
[9] Z. Shams and S. Seyedtabaii, Nonlinear ﬂexible link robotjoint-fault estimation using TS-fuzzy observers, Interna-tional Journal of Robotics and Automation, 35(1), 2020, doi:10.2316/J.2020.206-0214.
[10] Z. Geng, Study on the position control of electric cylinderbased on fuzzy adaptive PID, International Journal of Roboticsand Automation, 35(3), 2020, doi: 10.2316/J.2020.206-5226.
[11] Q. Zhou, H.Y. Li, and P. Shi, Decentralized adaptive fuzzytracking control for robot ﬁnger dynamics, IEEE Transactionson Fuzzy Systems, 23(3), 2015, 501–510.
[12] C. Ham, Z. Qu, and R. Johnson, Robust fuzzy control for robotmanipulators, IEE Proceedings-Control Theory Applications,147(2), 2000, 212–216.
[13] Y.Q. Wei, J.D. Zhang, L. Hou, et al., Backstepping adaptivefuzzy control for two-link robot manipulators, IJCSI Inter-national Journal of Computer Science Issues, 10(2), 2013,303–308.
[15]. Moreover, in these referencesand most similar researches, the manipulator drive systemis not considered. Although compared with the manipu-lator system itself, the dynamic characteristic of the drivesystem is relatively weak, it is necessary to consider thedrive system in the case of higher control requirements.It has been proved that there is a closed relationshipbetween coordination motion of human arm and musclecontraction
[16],
[17]. In fact, there is also the relationshipbetween the stable movements of robotic manipulators andthe drive systems. Thus, it is necessary to consider thedrive systems in designing the controller of robotic ma-nipulators. Recently, people have paid attention to thecombinations, but there is a little achievement. Shen
[18]considers the motor driving parameters mentioned, butthey are determined. The unknown motor driving pa-rameters are estimated by using neural network
[19], butit means that the uncertainties of the motor driving sys-tem can be parameterized like manipulator’s modelling1errors do. In fact, both the uncertainty of robotic ma-nipulator and the uncertainty of driving system are notclosely related. The stability performance of the roboticmanipulators is achievement considering the motor drivesystem
[20],
[21], but there are not uncertainties.To overcome the previous problems, this article de-signs a fuzzy adaptive controller based on feedback controltechnology and sliding mode theory on the basis of simul-taneously considering the modelling errors of manipulatorsystem, the external disturbances and the uncertainties ofdirect current (DC) motor drive system. The fuzzy systemadopted in the controller is used to estimate the uncer-tainty of the system modelling error, and the sliding modetechnology used compensates the estimated errors, the ex-ternal uncertainties and driving system uncertainties, thusensuring the global stability of the closed-loop system, andthe tracking error converging to zero gradually. The the-ory and simulations show that the designed controller canachieve good results.The main contribution of the overall article can besummarized as follows:1. Comparing with most existing researches, this articlenot only considers the manipulator system itself butalso considers the DC motor drive system.2. External disturbances, modelling uncertainties and un-certainties of DC motor drive system are taken intoaccount meanwhile in designing the controller.3. Two fuzzy logic systems are adopted to approximatethe modelling uncertainties to reduce the total numberof fuzzy rules.2. System ModelThe robotic manipulator is a kind of open chain structure,the freedom degrees of which are powered by independenttorques, respectively. Using the Lagrange equation, ann-link robotic manipulator system can be writtenas
[22]–[24]M(q)¨q + h(q, ˙q) = τ + τd (1)where q, ˙q, ¨q ∈ Rnare the joint displacement, velocity andacceleration, considering uncertainties, M(q) = M0(q) +ΔM(q) ∈ Rn×n[25], [26] is the inertia matrix, vectorh(q, ˙q) = h0(q, ˙q) + Δh(q, ˙q) ∈ Rnis coupled by the Coriolis,centrifugal and gravitational forces, τ ∈ Rnis the appliedjoint torque vector, τd ∈ Rnstands for the bounded uncer-tain external disturbances [27], [28], which are free from themanipulator itself. Here M0(q) and h0(q, ˙q) are nominalparts, ΔM(q) and Δh(q, ˙q) are uncertainties depending onthe manipulator itself in system matrices. Then (1) can berewritten asM0(q)¨q + h0(q, ˙q) = τ + τd + f(q, ˙q, ¨q) (2)where f(q, ˙q, ¨q) = −ΔM(q)¨q − Δh(q, ˙q) ∈ Rnlumps themodelling uncertainties.Without loss of generality, let the applied joint torquesbe provided by permanent-magnet DC motors. The DCmotors are usually manipulated by a servo ampliﬁerworking in current mode, then the electrical mathematicalexpression can be directly written as [19]τ = Kmim (3)where im ∈ Rnis the joint motor current vector and Km ∈Rn×nis a diagonal matrix composed of the motor torqueconstants. These constants are usually diﬃcult to measureor estimate, that is to say, Km is usually unknown. Whenthe servo ampliﬁers work in current mode, the operatingcurrent can be considered the same as the desired one,wherebyim = id = Ksau (4)where id ∈ Rnis the desired motor current vector,Ksa ∈ Rn×nis also a diagonal matrix composed of theservo ampliﬁer gains, which can be deﬁned by user and u isinput voltage of the servo ampliﬁer, which will be designedin the following text. Combining (3) and (4) shows the DCmotor output torque asτ = KmKsau = Ku (5)Then (2) can be changed asM0(q)¨q + h0(q, ˙q) = Ku + τd + f(q, ˙q, ¨q) (6)where K = KmKsa is also unknown. Like many referencesof this article, it is necessary here to give one property andtwo assumptions for (6).Property. The matrix M(q) is positive-deﬁnite andsatisﬁes M(q) ≤ m for some constant m > 0 andall q.Assumption 1. The nominal part M0(q) shows thesame property as M(q).Assumption 2. Each diagonal entry of K satisﬁeski > k0 > 0 with K0 = k0In being known, and letεg= K − K0 accompanied by εg≤ σg, where σgis aknown positive constant.Assumption 3. Each component of τd is less than aknown positive constant d0, that is to say, |τdi| ≤ d0.Assumptions 2 and 3 give the boundedness of theuncertainties K and τd. The lower bound K0 tells us thatthe ampliﬁer gains must not be too small and the upperbound d0 tells that the external disturbances cannot beinﬁnite. Nevertheless, Assumption 2 also tells that theampliﬁer gains must not be too lager. Obviously, the actualsituation of robotic manipulator entails these assumptions.3. Controller Design3.1 Preparatory WorksDeﬁnition. If all eigenvalues of a matrix V are in theopen left-half complex plane, then the matrix V is astable matrix.2Lemma 1 (Barbalat lemma) [29]. Assume thatx(t) ∈ [0, ∞] → R is ﬁrst-order continuously diﬀeren-tiable and there exists a limit when t → ∞, if ˙x(t) isuniformly continuous, then limt→∞ ˙x(t) = 0.Lemma 2 [30]. If a, c ∈ Rnand D ∈ Rn×n, then there isaTDc = tr(DTacT).3.2 Nominal SystemWhen there is no any uncertainties in (6), it means thatf(q, ˙q, ¨q) = 0, τd = 0 and K is known very well, (6)becomesM0(q)¨q + h0(q, ˙q) = Ku (7)which is called as the nominal system. We can easily usethe feedback control theory to design u as follows:u = K−1[M0(q)(¨qd − K1 ˙e − K2e) + h0(q, ˙q)] (8)where qd, ˙qd, ¨qd ∈ Rnare given joint displacement, velocityand acceleration, e = q − qd deﬁnes the tracking error,K1, K2 ∈ Rn×nare positive deﬁnite diagonal matricesselected by user. Substituting (8) into (7), we obtain thefollowing:¨e + K1 ˙e + K2e = 0 (9)If S = [eT˙eT]Tis deﬁned, (9) is equivalent to thefollowing linear state equation:˙S = AS (10)whereA =⎡⎣0n In−K2 −K1⎤⎦is a stable matrix. In fact, we can choose the followingelementary matrixU =⎡⎣In 0n−K−11 K2 In⎤⎦to apply the congruent transformation to A, obtainingUTAU =⎡⎣−K−11 K2 In + K20n −K1⎤⎦Obviously, all eigenvalues of a matrix UTAU are in theopen left-half complex plane, and then all eigenvalues ofthe matrix A are in the open left-half complex plane too.That is to say, A is a stable matrix. Therefore, (10) isasymptotically stable and e, ˙e will converge to zero.In addition, for a stable matrix A and any given posi-tive deﬁnite matrix Q, there is always some positive deﬁnitematrix P satisfying the following Lyapunov equation:PA + ATP = −Q (11)where Q is usually chosen as the identity matrix.3.3 Uncertain SystemConsidering the uncertainties of K, τd and f(q, ˙q, ¨q) in (6),the control input u in (8) does not guarantee e, ˙e convergeto zero; hence, we need redesign u. For the sake of usingthe feedback control theory, the parameter uncertaintyf(q, ˙q, ¨q) needs to appear in the form of its estimationˆf(q, ˙q, ¨q) and K is replaced by its lower bound K0 in thedesigned controller u. Then the following modiﬁed controlinput will be obtained in a relatively direct way:u=u1 =K−10 M0(q)(¨qd −K1 ˙e − K2e)+h0(q, ˙q)− ˆf(q, ˙q, ¨q)(12)Substituting it into (6), obtain the following errordynamics:¨e + K1 ˙e + K2e = (f (q, ˙q, ¨q) − ˆf (q, ˙q, ¨q)) + (K − K0)u1 + τd(13)Because of physical implementation constraint, the es-timation ˆf(q, ˙q, ¨q) cannot completely equal to the unknownbut true value f(q, ˙q, ¨q). Coupled with the non-negligibleτd and K − K0 = 0, it is diﬃcult to guarantee the sta-bility and convergence of the error dynamics (13). Foreliminating the eﬀects of approximation errors and otheruncertainties, we need to redesign the control input u againby adding another term u2, whereupon the control input ubecomes as follows:u = u1 + u2 (14)As it is proved that the fuzzy logic has many advan-tages, in the following the uncertainty f(q, ˙q, ¨q) can beestimated by adopting the fuzzy theory [31]–[33].3.4 Fuzzy ApproximatorsThis subsection will take the trouble to introduce the fuzzyapproximators as some references do to provide a trouble-free design process. It is very meaningful to assume anuncertain system as y = f(ω). ω ∈ Rnis an input vector,from which a mapping is performed to an output variabley ∈ R by employing the fuzzy IF-THEN rules in the fuzzyinference engine. The ith fuzzy rule is deﬁned asRi: If ω1 is μi1 and · · · and ωn is μin, then y is Biwhere μi1, . . . , μin and Biare fuzzy sets deﬁned by fuzzymembership functions. When the singleton fuzziﬁer, theproduct-inference engine and the centre-average defuzziﬁerare used, the variable y can be deﬁned asy =Ni=1 ˜yi nj=1 μij(ωj)Ni=1nj=1 μij(ωj)= θTψ(ω) (15)where N is the total number of fuzzy rules, θ =˜y1, . . . , ˜yN Tis an adjustable parameter vector with each˜yibeing the point at which the fuzzy membership function3shows the maximum, and ψ(ω) = [ψ1, . . . , ψN ]Tis a fuzzybasis vector whose lth element is of the following form:ψl =nj=1 μij(ωj)Ni=1nj=1 μij(ωj)(16)It is constructive to divide the uncertainty f (q, ˙q, ¨q)into an addition of two functions before applying therecommended fuzzy systems (15), thereby reducing thenumber of fuzzy rules. The addition equation is of the fol-lowing form:f(q, ˙q, ¨q) = f1(q, ˙q) + f2(q, ¨q) (17)where f1(q, ˙q) = −Δh(q, ˙q) and f2(q, ¨q) = −ΔM(q)¨q.Then these two functions can be approximated by theintroduced fuzzy systems (15), and their approximationsexpress, respectively, as follows:ˆf1(q, ˙q|θ1) = θ1T1 F1(q, ˙q), . . . , θ1Tn F1(q, ˙q)T= θ1TF1(q, ˙q)ˆf2(q, ¨q|θ2) = θ2T1 F2(q, ¨q), . . . , θ2Tn F2(q, ¨q)T= θ2TF2(q, ¨q)(18)The total approximation ˆf(q, ˙q, ¨q) is gotten after sum-ming ˆf1(q, ˙q|θ1) and ˆf2(q, ¨q|θ2). Here the minimum ap-proximation errors ε1and ε2also need to be given byε1= f1(q, ˙q) − θ1∗TF1(q, ˙q)ε2= f2(q, ¨q) − θ2∗TF2(q, ¨q)(19)where θ1∗and θ2∗are the optimal parameter matrices ofeach fuzzy logic system, respectively. ε1and ε2satisfythe following assumption according to the approximationprinciple.Assumption 4. For some known positive constants σ1and σ2, there areε1i ≤ σ1, ε2i ≤ σ2This subsection will be end after giving the followingapproximation errors of weight matrices:˜θ1= θ1− θ1∗, ˜θ2= θ2− θ2∗and their derivatives are˙˜θ1= ˙θ1and˙˜θ2= ˙θ2, respectively.3.5 Stability AnalysisFor designing a well-deﬁned adaptive fuzzy controller con-sidering the eﬀects of the uncertainties in DC motor, theapproximation errors and the external disturbances, thecompensation term u2 in (14) is developed asu2 = −λsgn(δ) (20)Deﬁne δ = [P21 P22] S as the ﬁltered errors, whereP21 ∈ Rnand P22 ∈ Rnare parts of the positive deﬁnitematrix P in (11), which is of the following form:P =⎡⎣P11 P21P21 P22⎤⎦ (21)λ is a gain matrix deﬁned by the following equation:λ = k−10 (σ1+ σ2+ u1 σg+ d0)In (22)and sgn(δ) ∈ Rnis a symbolic function vector made up bythe following n-scalar symbolic functions:sgni(δi) =⎧⎪⎪⎪⎨⎪⎪⎪⎩1 δi > 00 δi = 0−1 δi < 0,(23)Combining the actual manipulator system (6) with thecontroller (14) accompanied by (12) and (20)–(23), we getthe following ﬁnal error linear closed-loop state equation:˙S = AS + B (24)where B = [0TbT]Twith b = −˜θ1TF1(q, ˙q)−˜θ2TF2(q, ¨q)+εgu1 +Ku2 +ε1+ε2+τd, A and S are the same as in (10).Then, the stability of the closed-loop system (24) can bedescribed by the following theorem.Theorem. Considering Assumptions 1–4, the ﬁnallinear closed-loop state equation (24) obtained from(6), (12), (14) and (20)–(23) is asymptotically stable.Proof: Choose the weight matrix updating algo-rithms as˙θ1= η1F1(q, ˙q)δT˙θ2= η2F2(q, ¨q)δT(25)where η1 and η2 are positive constants.Then construct the following Lyapunov function:V =12STPS +12tr ˜θ1Tη−11˜θ1+12tr ˜θ2Tη−12˜θ2(26)By diﬀerentiating it with respect to time along theclosed-loop state (24), considering the aforementionedLyapunov equation (11) and its solution (21), plus thedeﬁnition δ = [P21 P22]S, we get the following:˙V = −12STQS − F1T(q, ˙q)˜θ1δ − F2T(q, ¨q)˜θ2δ+ εgu1 + Ku2 + ε1+ ε2+ τdTδ+ tr ˜θT1 η−11˙˜θ1 + tr ˜θT2 η−12˙˜θ2Then, considering (25) and using Lemma 2, we get thefollowing:˙V = −12STQS + εgu1 + Ku2 + ε1+ ε2+ τdTδ4Further, retrospecting (20)–(22) and aforementionedassumptions, ˙V becomes as follows:V = −12STQS + εgu1 − k−10 (σ1+ σ2+ u1 σg+ d0)Ksgn(δ)+ ε1+ ε2+ τdTδFurthermore, after using (23), ˙V becomes again asfollows:˙V = −12STQS + εgu1 − k−10 (σ1+ σ2+ u1 σg+ d0)[k1sgn1(δ1), . . . , knsgnn(δn)]T+ ε1+ ε2+ τdTδConsidering (12), for the ith element of the secondpart of ˙V , when δi ≥ 0, there isεgi u1i − kik−10 (σ1+ σ2+ u1 σg+ d0) + ε1i + ε2i + τdi ≤ 0and when δi ≤ 0, there is also the following:−(εgi u1i +kik−10 (σ1+σ2+ u1 σg+d0)+ε1i +ε2i +τdi) ≤ 0In other words, the second part of ˙V is never greaterthan zero. Then, there is always˙V ≤ −12STQS ≤ 0and limt→∞ S = 0. In fact, if we let V1(t) = V (t) −t0(V + (1/2)STQS)dτ, it is easy to prove ˙V1 = −1/2STQSuniform and continuous [30]. Using the Lemma 1, we canconclude limt→∞˙V1 = 0, whereupon S → 0 as t → ∞.So, the linear closed-loop error state equation (24) isasymptotically stable.The architecture of the proposed control scheme isshown in Fig. 1.Figure 1. Architecture of the proposed control scheme.Figure 2. Two-link robotic manipulator actuated by DCmotors.Table 1Nominal Values and Limits of Uncertaintiesm10 1 kg Δm1 0.1 kgm20 1 kg Δm2 0.1 kg4. Simulation ResultsTo verify the feasibility of the designed controller, a two-link manipulator is taken to give the simulation results.The conﬁguration of the manipulator actuated by DCmotors is shown in Fig. 2 [34], [35].The manipulator model entries in (1) are described byM(q) =⎡⎣M11 M12M21 M22⎤⎦, h(q, ˙q) =⎡⎣h1h2⎤⎦, τd =⎡⎣0.5 cos(t)0.5 sin(t)⎤⎦where M11 = (m1 + m2)l21, M22 = m2l22, M12 = M21 =m2l1l2 cos(q1 − q2), h1 = m2l1l2 sin(q1 − q2) ˙q22 − (m1 +m2)l1 sin(q1)g, h2 = m2l1l2 sin(q1 − q2) ˙q21 − m2l2 sin(q2)g.Without the loss of generality, the modelling uncer-tainties are only embodied in m1 and m2. Then the nom-inal values of m1, m2 and their limits of uncertainties areshown in Table 1.5Figure 3. Simulation results when the errors are 0.1 kg: (a) position tracking of joints and (b) control input of joints.Figure 4. Simulation results when the errors are 0.3 kg: (a) position tracking of joints with K1 = 30I2, K2 = 5I2, (b) controlinput of joints with K1 = 30I2, K2 = 5I2, (c) position tracking of joints with K1 = 50I2, K2 = 10I2 and (d) control input ofjoints with K1 = 50I2, K2 = 10I2.The initial angles and their derivatives are selectedas q1(0) = q2(0) = 0, ˙q1(0) = ˙q2(0) = 0, respectively, the de-sired reference trajectories are chosen as qd1 = sin t andqd2 = 2 sin t. Another parameters are K1 = 30I2, K2 = 5I2,σ1= σ2= σg= 1, η1 = η2 = 0.1, d0 = 0.5, Q = I4, l1 = l2 = 1,k1 = k2 = 10.5, k0 = 10, g = 9.8.In the simulation, we adopt ﬁve fuzzy levels, i.e., NB,NS, ZO, PS, PB on the universe of each input variable anduse the following Gaussian membership function:μAli(xi) = exp −xi − ¯xliπ/242where ¯xli are −π/6, −π/12, 0, π/12, and π/6, respectively.The simulation results are shown in Fig. 3. Part (a) of Fig. 3is the trajectory tracking and part (b) is control input.When uncertainties increase, such as the upper limitsof quality errors here are 0.3 kg, but other parameters haveno changes, the simulation results are shown in Fig. 4(a)6Figure 5. Simulation results when σg= 5: (a) position tracking of joints and (b) control input of joints.and (b). From Fig. 4(a), we can see the tracking errorsincrease signiﬁcantly. However, if we enhance the gainmatrices K1 and K2, the tracking errors will reduce again.Parts (c) and (d) of Fig. 4 are simulation results withK1 = 50I2, K2 = 10I2.When the uncertainties of motor drive parameters in-crease, i.e., σgis bigger, but other parameters have nochanges, the tracking errors are aﬀected slightly. Never-theless, the volatility of control input will increase signiﬁ-cantly. Parts (a) and (b) of Fig. 5 are control input resultswhen the quality errors are 0.1 and 0.3 kg, respectively, un-der the condition of K1 = 30I2, K2 = 5I2 and σg= 5. Theseﬁgures tell us in the practical application, it is necessaryto get more accurate estimated values of the motor drivingparameters.5. ConclusionConsidering the DC motor drive system, an adaptive fuzzycontroller is designed for the tracking control problemof manipulator. First, the robotic manipulator systemcombines the electrical mathematical system to get themodel that is discussed in this article. Second, the feedbackcontrol technique is used to transform the model obtainedearlier into a concise state equation. Then, based onthe linear control method, the fuzzy control combinesthe adaptive sliding mode technique to get the proposedadaptive fuzzy controller. Last, the eﬀectiveness is notonly proved by using the Lyapunov stability theory butalso shown by taking a two-link manipulator as an exampleof simulation.Compared with most existing research results, thisarticle not only considers the modelling errors and externalinterferences of the manipulator system but also considersthe uncertainties of the motor drive system, thus makingour research results closer to the actual operating systemof the manipulator. Although the designed controller is tosolve the tracking problem of manipulator system in thisarticle, this design method is also suitable for many multi-input and multi-output systems which possess the similardynamic characteristics.References[1] K.J. Cheng and P.Y. Cheng, Investigation on velocity perfor-mance deviation of serial manipulators resulted from fabrica-tion errors, International Journal of Robotics and Automation,32(4), 2017, 324–332.[2] Z.X. Yang and D. Zhang, Energy optimal adaption and mo-tion planning of a 3-RRS balanced manipulator, Interna-tional Journal of Robotics and Automation, 34(5), 2019, doi:10.2316/J.2019.206-0171.[3] D. Cafolla, M. Wang, G. Carbone, et al., LARMbot: A newhumanoid robot with parallel mechanisms, Proc. 21st CISM-IFToMM Symposium, Udine, Italy, 2016, 275–283.[4] M. Russo, D. Cafolla and M. Ceccarelli, Development ofLARMbot 2, a novel humanoid robot with parallel architec-tures, Proc. 4th IFToMM Symposium on Mechanism Designfor Robotics, Udine, Italy, 2018, 17–24.[5] J. Chen and H. Lau, Policy gradient-based inverse kinematicsreﬁnement for tendon-driven serpentine surgical manipulator,International Journal of Robotics and Automation, 34(3), 2019,doi: 10.2316/J.2019.206-5461.[6] F. Yang, G.L. Zhang, L. Yuan, et al., End-eﬀector optimaltracking control of free-ﬂoating space robot, Journal of Astro-nautics, 37(7), 2016, 846–853.[7] J.Y. Li, L. Wan, H. 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Johnson, Robust fuzzy control for robotmanipulators, IEE Proceedings-Control Theory Applications,147(2), 2000, 212–216.[13] Y.Q. Wei, J.D. Zhang, L. Hou, et al., Backstepping adaptivefuzzy control for two-link robot manipulators, IJCSI Inter-national Journal of Computer Science Issues, 10(2), 2013,303–308.[14] M.M. Azimi and H.R. Kooﬁgar, Adaptive fuzzy backsteppingcontroller design for uncertain underactuated robotic systems,Nonlinear Dynamics, 79(2), 2015, 1–12.7[15] Z.J. Yang and H.G. Zhang, A fuzzy adaptive tracking controlfor a class of uncertain strick-feedback nonlinear systems withdead-zone input, Neurocomputing, 272, 2018, 130–135.[16] D.A. Gabriel, Shoulder and elbow muscle activity in goal-directed arm movements, Experimental Brain Research, 116(2),1997, 359–366.[17] N.T. Antony and P.J. Keir, Eﬀects of posture, movement andhand load on shoulder muscle activity, Journal of Electromyo-graphy and Kinesiology, 20(2), 2010, 191–198.[18] T.L. Shen, Robust control fundamentals of robot (Beijing:Tsinghua University Press, 2000).[19] Y.X. Wu and C. Wang, Adaptive neural network control andleaning for uncertain robot, Control Theory & Applications,30(8), 2013, 990–997.[20] S. Malagari and B.J. Driessen, Globally exponential con-troller/observer for tracking in robots with DC motor dynamicsand only link position measurement, International Journal ofModelling, Identiﬁcation and Control, 19(1), 2013, 1–12.[21] V.M. Hernandez-Guzman and J. Orrante-Sakanassi, GlobalPID control of robot manipulators equipped with PMSMS,Asian Journal of Control, 20(1), 2018, 236–249.[22] Z.H. Man and M. Palaniswami, Robust tracking control forrigid robotic manipulators, IEEE Transactions on AutomaticControl, 39(1), 1994, 154–159.
[24]M(q)¨q + h(q, ˙q) = τ + τd (1)where q, ˙q, ¨q ∈ Rnare the joint displacement, velocity andacceleration, considering uncertainties, M(q) = M0(q) +ΔM(q) ∈ Rn×n
[25],
[26] is the inertia matrix, vectorh(q, ˙q) = h0(q, ˙q) + Δh(q, ˙q) ∈ Rnis coupled by the Coriolis,centrifugal and gravitational forces, τ ∈ Rnis the appliedjoint torque vector, τd ∈ Rnstands for the bounded uncer-tain external disturbances
[27],
[28], which are free from themanipulator itself. Here M0(q) and h0(q, ˙q) are nominalparts, ΔM(q) and Δh(q, ˙q) are uncertainties depending onthe manipulator itself in system matrices. Then (1) can berewritten asM0(q)¨q + h0(q, ˙q) = τ + τd + f(q, ˙q, ¨q) (2)where f(q, ˙q, ¨q) = −ΔM(q)¨q − Δh(q, ˙q) ∈ Rnlumps themodelling uncertainties.Without loss of generality, let the applied joint torquesbe provided by permanent-magnet DC motors. The DCmotors are usually manipulated by a servo ampliﬁerworking in current mode, then the electrical mathematicalexpression can be directly written as [19]τ = Kmim (3)where im ∈ Rnis the joint motor current vector and Km ∈Rn×nis a diagonal matrix composed of the motor torqueconstants. These constants are usually diﬃcult to measureor estimate, that is to say, Km is usually unknown. Whenthe servo ampliﬁers work in current mode, the operatingcurrent can be considered the same as the desired one,wherebyim = id = Ksau (4)where id ∈ Rnis the desired motor current vector,Ksa ∈ Rn×nis also a diagonal matrix composed of theservo ampliﬁer gains, which can be deﬁned by user and u isinput voltage of the servo ampliﬁer, which will be designedin the following text. Combining (3) and (4) shows the DCmotor output torque asτ = KmKsau = Ku (5)Then (2) can be changed asM0(q)¨q + h0(q, ˙q) = Ku + τd + f(q, ˙q, ¨q) (6)where K = KmKsa is also unknown. Like many referencesof this article, it is necessary here to give one property andtwo assumptions for (6).Property. The matrix M(q) is positive-deﬁnite andsatisﬁes M(q) ≤ m for some constant m > 0 andall q.Assumption 1. The nominal part M0(q) shows thesame property as M(q).Assumption 2. Each diagonal entry of K satisﬁeski > k0 > 0 with K0 = k0In being known, and letεg= K − K0 accompanied by εg≤ σg, where σgis aknown positive constant.Assumption 3. Each component of τd is less than aknown positive constant d0, that is to say, |τdi| ≤ d0.Assumptions 2 and 3 give the boundedness of theuncertainties K and τd. The lower bound K0 tells us thatthe ampliﬁer gains must not be too small and the upperbound d0 tells that the external disturbances cannot beinﬁnite. Nevertheless, Assumption 2 also tells that theampliﬁer gains must not be too lager. Obviously, the actualsituation of robotic manipulator entails these assumptions.3. Controller Design3.1 Preparatory WorksDeﬁnition. If all eigenvalues of a matrix V are in theopen left-half complex plane, then the matrix V is astable matrix.2Lemma 1 (Barbalat lemma)
[29]. Assume thatx(t) ∈ [0, ∞] → R is ﬁrst-order continuously diﬀeren-tiable and there exists a limit when t → ∞, if ˙x(t) isuniformly continuous, then limt→∞ ˙x(t) = 0.Lemma 2
[30]. If a, c ∈ Rnand D ∈ Rn×n, then there isaTDc = tr(DTacT).3.2 Nominal SystemWhen there is no any uncertainties in (6), it means thatf(q, ˙q, ¨q) = 0, τd = 0 and K is known very well, (6)becomesM0(q)¨q + h0(q, ˙q) = Ku (7)which is called as the nominal system. We can easily usethe feedback control theory to design u as follows:u = K−1[M0(q)(¨qd − K1 ˙e − K2e) + h0(q, ˙q)] (8)where qd, ˙qd, ¨qd ∈ Rnare given joint displacement, velocityand acceleration, e = q − qd deﬁnes the tracking error,K1, K2 ∈ Rn×nare positive deﬁnite diagonal matricesselected by user. Substituting (8) into (7), we obtain thefollowing:¨e + K1 ˙e + K2e = 0 (9)If S = [eT˙eT]Tis deﬁned, (9) is equivalent to thefollowing linear state equation:˙S = AS (10)whereA =⎡⎣0n In−K2 −K1⎤⎦is a stable matrix. In fact, we can choose the followingelementary matrixU =⎡⎣In 0n−K−11 K2 In⎤⎦to apply the congruent transformation to A, obtainingUTAU =⎡⎣−K−11 K2 In + K20n −K1⎤⎦Obviously, all eigenvalues of a matrix UTAU are in theopen left-half complex plane, and then all eigenvalues ofthe matrix A are in the open left-half complex plane too.That is to say, A is a stable matrix. Therefore, (10) isasymptotically stable and e, ˙e will converge to zero.In addition, for a stable matrix A and any given posi-tive deﬁnite matrix Q, there is always some positive deﬁnitematrix P satisfying the following Lyapunov equation:PA + ATP = −Q (11)where Q is usually chosen as the identity matrix.3.3 Uncertain SystemConsidering the uncertainties of K, τd and f(q, ˙q, ¨q) in (6),the control input u in (8) does not guarantee e, ˙e convergeto zero; hence, we need redesign u. For the sake of usingthe feedback control theory, the parameter uncertaintyf(q, ˙q, ¨q) needs to appear in the form of its estimationˆf(q, ˙q, ¨q) and K is replaced by its lower bound K0 in thedesigned controller u. Then the following modiﬁed controlinput will be obtained in a relatively direct way:u=u1 =K−10 M0(q)(¨qd −K1 ˙e − K2e)+h0(q, ˙q)− ˆf(q, ˙q, ¨q)(12)Substituting it into (6), obtain the following errordynamics:¨e + K1 ˙e + K2e = (f (q, ˙q, ¨q) − ˆf (q, ˙q, ¨q)) + (K − K0)u1 + τd(13)Because of physical implementation constraint, the es-timation ˆf(q, ˙q, ¨q) cannot completely equal to the unknownbut true value f(q, ˙q, ¨q). Coupled with the non-negligibleτd and K − K0 = 0, it is diﬃcult to guarantee the sta-bility and convergence of the error dynamics (13). Foreliminating the eﬀects of approximation errors and otheruncertainties, we need to redesign the control input u againby adding another term u2, whereupon the control input ubecomes as follows:u = u1 + u2 (14)As it is proved that the fuzzy logic has many advan-tages, in the following the uncertainty f(q, ˙q, ¨q) can beestimated by adopting the fuzzy theory
[31]–[33].3.4 Fuzzy ApproximatorsThis subsection will take the trouble to introduce the fuzzyapproximators as some references do to provide a trouble-free design process. It is very meaningful to assume anuncertain system as y = f(ω). ω ∈ Rnis an input vector,from which a mapping is performed to an output variabley ∈ R by employing the fuzzy IF-THEN rules in the fuzzyinference engine. The ith fuzzy rule is deﬁned asRi: If ω1 is μi1 and · · · and ωn is μin, then y is Biwhere μi1, . . . , μin and Biare fuzzy sets deﬁned by fuzzymembership functions. When the singleton fuzziﬁer, theproduct-inference engine and the centre-average defuzziﬁerare used, the variable y can be deﬁned asy =Ni=1 ˜yi nj=1 μij(ωj)Ni=1nj=1 μij(ωj)= θTψ(ω) (15)where N is the total number of fuzzy rules, θ =˜y1, . . . , ˜yN Tis an adjustable parameter vector with each˜yibeing the point at which the fuzzy membership function3shows the maximum, and ψ(ω) = [ψ1, . . . , ψN ]Tis a fuzzybasis vector whose lth element is of the following form:ψl =nj=1 μij(ωj)Ni=1nj=1 μij(ωj)(16)It is constructive to divide the uncertainty f (q, ˙q, ¨q)into an addition of two functions before applying therecommended fuzzy systems (15), thereby reducing thenumber of fuzzy rules. The addition equation is of the fol-lowing form:f(q, ˙q, ¨q) = f1(q, ˙q) + f2(q, ¨q) (17)where f1(q, ˙q) = −Δh(q, ˙q) and f2(q, ¨q) = −ΔM(q)¨q.Then these two functions can be approximated by theintroduced fuzzy systems (15), and their approximationsexpress, respectively, as follows:ˆf1(q, ˙q|θ1) = θ1T1 F1(q, ˙q), . . . , θ1Tn F1(q, ˙q)T= θ1TF1(q, ˙q)ˆf2(q, ¨q|θ2) = θ2T1 F2(q, ¨q), . . . , θ2Tn F2(q, ¨q)T= θ2TF2(q, ¨q)(18)The total approximation ˆf(q, ˙q, ¨q) is gotten after sum-ming ˆf1(q, ˙q|θ1) and ˆf2(q, ¨q|θ2). Here the minimum ap-proximation errors ε1and ε2also need to be given byε1= f1(q, ˙q) − θ1∗TF1(q, ˙q)ε2= f2(q, ¨q) − θ2∗TF2(q, ¨q)(19)where θ1∗and θ2∗are the optimal parameter matrices ofeach fuzzy logic system, respectively. ε1and ε2satisfythe following assumption according to the approximationprinciple.Assumption 4. For some known positive constants σ1and σ2, there areε1i ≤ σ1, ε2i ≤ σ2This subsection will be end after giving the followingapproximation errors of weight matrices:˜θ1= θ1− θ1∗, ˜θ2= θ2− θ2∗and their derivatives are˙˜θ1= ˙θ1and˙˜θ2= ˙θ2, respectively.3.5 Stability AnalysisFor designing a well-deﬁned adaptive fuzzy controller con-sidering the eﬀects of the uncertainties in DC motor, theapproximation errors and the external disturbances, thecompensation term u2 in (14) is developed asu2 = −λsgn(δ) (20)Deﬁne δ = [P21 P22] S as the ﬁltered errors, whereP21 ∈ Rnand P22 ∈ Rnare parts of the positive deﬁnitematrix P in (11), which is of the following form:P =⎡⎣P11 P21P21 P22⎤⎦ (21)λ is a gain matrix deﬁned by the following equation:λ = k−10 (σ1+ σ2+ u1 σg+ d0)In (22)and sgn(δ) ∈ Rnis a symbolic function vector made up bythe following n-scalar symbolic functions:sgni(δi) =⎧⎪⎪⎪⎨⎪⎪⎪⎩1 δi > 00 δi = 0−1 δi < 0,(23)Combining the actual manipulator system (6) with thecontroller (14) accompanied by (12) and (20)–(23), we getthe following ﬁnal error linear closed-loop state equation:˙S = AS + B (24)where B = [0TbT]Twith b = −˜θ1TF1(q, ˙q)−˜θ2TF2(q, ¨q)+εgu1 +Ku2 +ε1+ε2+τd, A and S are the same as in (10).Then, the stability of the closed-loop system (24) can bedescribed by the following theorem.Theorem. Considering Assumptions 1–4, the ﬁnallinear closed-loop state equation (24) obtained from(6), (12), (14) and (20)–(23) is asymptotically stable.Proof: Choose the weight matrix updating algo-rithms as˙θ1= η1F1(q, ˙q)δT˙θ2= η2F2(q, ¨q)δT(25)where η1 and η2 are positive constants.Then construct the following Lyapunov function:V =12STPS +12tr ˜θ1Tη−11˜θ1+12tr ˜θ2Tη−12˜θ2(26)By diﬀerentiating it with respect to time along theclosed-loop state (24), considering the aforementionedLyapunov equation (11) and its solution (21), plus thedeﬁnition δ = [P21 P22]S, we get the following:˙V = −12STQS − F1T(q, ˙q)˜θ1δ − F2T(q, ¨q)˜θ2δ+ εgu1 + Ku2 + ε1+ ε2+ τdTδ+ tr ˜θT1 η−11˙˜θ1 + tr ˜θT2 η−12˙˜θ2Then, considering (25) and using Lemma 2, we get thefollowing:˙V = −12STQS + εgu1 + Ku2 + ε1+ ε2+ τdTδ4Further, retrospecting (20)–(22) and aforementionedassumptions, ˙V becomes as follows:V = −12STQS + εgu1 − k−10 (σ1+ σ2+ u1 σg+ d0)Ksgn(δ)+ ε1+ ε2+ τdTδFurthermore, after using (23), ˙V becomes again asfollows:˙V = −12STQS + εgu1 − k−10 (σ1+ σ2+ u1 σg+ d0)[k1sgn1(δ1), . . . , knsgnn(δn)]T+ ε1+ ε2+ τdTδConsidering (12), for the ith element of the secondpart of ˙V , when δi ≥ 0, there isεgi u1i − kik−10 (σ1+ σ2+ u1 σg+ d0) + ε1i + ε2i + τdi ≤ 0and when δi ≤ 0, there is also the following:−(εgi u1i +kik−10 (σ1+σ2+ u1 σg+d0)+ε1i +ε2i +τdi) ≤ 0In other words, the second part of ˙V is never greaterthan zero. Then, there is always˙V ≤ −12STQS ≤ 0and limt→∞ S = 0. In fact, if we let V1(t) = V (t) −t0(V + (1/2)STQS)dτ, it is easy to prove ˙V1 = −1/2STQSuniform and continuous [30]. Using the Lemma 1, we canconclude limt→∞˙V1 = 0, whereupon S → 0 as t → ∞.So, the linear closed-loop error state equation (24) isasymptotically stable.The architecture of the proposed control scheme isshown in Fig. 1.Figure 1. Architecture of the proposed control scheme.Figure 2. Two-link robotic manipulator actuated by DCmotors.Table 1Nominal Values and Limits of Uncertaintiesm10 1 kg Δm1 0.1 kgm20 1 kg Δm2 0.1 kg4. Simulation ResultsTo verify the feasibility of the designed controller, a two-link manipulator is taken to give the simulation results.The conﬁguration of the manipulator actuated by DCmotors is shown in Fig. 2 [34], [35].The manipulator model entries in (1) are described byM(q) =⎡⎣M11 M12M21 M22⎤⎦, h(q, ˙q) =⎡⎣h1h2⎤⎦, τd =⎡⎣0.5 cos(t)0.5 sin(t)⎤⎦where M11 = (m1 + m2)l21, M22 = m2l22, M12 = M21 =m2l1l2 cos(q1 − q2), h1 = m2l1l2 sin(q1 − q2) ˙q22 − (m1 +m2)l1 sin(q1)g, h2 = m2l1l2 sin(q1 − q2) ˙q21 − m2l2 sin(q2)g.Without the loss of generality, the modelling uncer-tainties are only embodied in m1 and m2. Then the nom-inal values of m1, m2 and their limits of uncertainties areshown in Table 1.5Figure 3. Simulation results when the errors are 0.1 kg: (a) position tracking of joints and (b) control input of joints.Figure 4. Simulation results when the errors are 0.3 kg: (a) position tracking of joints with K1 = 30I2, K2 = 5I2, (b) controlinput of joints with K1 = 30I2, K2 = 5I2, (c) position tracking of joints with K1 = 50I2, K2 = 10I2 and (d) control input ofjoints with K1 = 50I2, K2 = 10I2.The initial angles and their derivatives are selectedas q1(0) = q2(0) = 0, ˙q1(0) = ˙q2(0) = 0, respectively, the de-sired reference trajectories are chosen as qd1 = sin t andqd2 = 2 sin t. Another parameters are K1 = 30I2, K2 = 5I2,σ1= σ2= σg= 1, η1 = η2 = 0.1, d0 = 0.5, Q = I4, l1 = l2 = 1,k1 = k2 = 10.5, k0 = 10, g = 9.8.In the simulation, we adopt ﬁve fuzzy levels, i.e., NB,NS, ZO, PS, PB on the universe of each input variable anduse the following Gaussian membership function:μAli(xi) = exp −xi − ¯xliπ/242where ¯xli are −π/6, −π/12, 0, π/12, and π/6, respectively.The simulation results are shown in Fig. 3. Part (a) of Fig. 3is the trajectory tracking and part (b) is control input.When uncertainties increase, such as the upper limitsof quality errors here are 0.3 kg, but other parameters haveno changes, the simulation results are shown in Fig. 4(a)6Figure 5. Simulation results when σg= 5: (a) position tracking of joints and (b) control input of joints.and (b). From Fig. 4(a), we can see the tracking errorsincrease signiﬁcantly. However, if we enhance the gainmatrices K1 and K2, the tracking errors will reduce again.Parts (c) and (d) of Fig. 4 are simulation results withK1 = 50I2, K2 = 10I2.When the uncertainties of motor drive parameters in-crease, i.e., σgis bigger, but other parameters have nochanges, the tracking errors are aﬀected slightly. Never-theless, the volatility of control input will increase signiﬁ-cantly. Parts (a) and (b) of Fig. 5 are control input resultswhen the quality errors are 0.1 and 0.3 kg, respectively, un-der the condition of K1 = 30I2, K2 = 5I2 and σg= 5. Theseﬁgures tell us in the practical application, it is necessaryto get more accurate estimated values of the motor drivingparameters.5. ConclusionConsidering the DC motor drive system, an adaptive fuzzycontroller is designed for the tracking control problemof manipulator. First, the robotic manipulator systemcombines the electrical mathematical system to get themodel that is discussed in this article. Second, the feedbackcontrol technique is used to transform the model obtainedearlier into a concise state equation. Then, based onthe linear control method, the fuzzy control combinesthe adaptive sliding mode technique to get the proposedadaptive fuzzy controller. Last, the eﬀectiveness is notonly proved by using the Lyapunov stability theory butalso shown by taking a two-link manipulator as an exampleof simulation.Compared with most existing research results, thisarticle not only considers the modelling errors and externalinterferences of the manipulator system but also considersthe uncertainties of the motor drive system, thus makingour research results closer to the actual operating systemof the manipulator. Although the designed controller is tosolve the tracking problem of manipulator system in thisarticle, this design method is also suitable for many multi-input and multi-output systems which possess the similardynamic characteristics.References[1] K.J. Cheng and P.Y. Cheng, Investigation on velocity perfor-mance deviation of serial manipulators resulted from fabrica-tion errors, International Journal of Robotics and Automation,32(4), 2017, 324–332.[2] Z.X. Yang and D. Zhang, Energy optimal adaption and mo-tion planning of a 3-RRS balanced manipulator, Interna-tional Journal of Robotics and Automation, 34(5), 2019, doi:10.2316/J.2019.206-0171.[3] D. Cafolla, M. Wang, G. Carbone, et al., LARMbot: A newhumanoid robot with parallel mechanisms, Proc. 21st CISM-IFToMM Symposium, Udine, Italy, 2016, 275–283.[4] M. Russo, D. Cafolla and M. Ceccarelli, Development ofLARMbot 2, a novel humanoid robot with parallel architec-tures, Proc. 4th IFToMM Symposium on Mechanism Designfor Robotics, Udine, Italy, 2018, 17–24.[5] J. Chen and H. Lau, Policy gradient-based inverse kinematicsreﬁnement for tendon-driven serpentine surgical manipulator,International Journal of Robotics and Automation, 34(3), 2019,doi: 10.2316/J.2019.206-5461.[6] F. Yang, G.L. Zhang, L. Yuan, et al., End-eﬀector optimaltracking control of free-ﬂoating space robot, Journal of Astro-nautics, 37(7), 2016, 846–853.[7] J.Y. Li, L. Wan, H. Huang, et al., Adaptive control methodof underwater vehicle manipulator system under disturbances,Journal of Tianjin University (Science and Technology), 51(4),2018, 413–421.[8] C.J. Huang, U. Farooq, H.Y. Liu, et al., A PSO-tuned fuzzylogic system for position tracking of mobile robot, Interna-tional Journal of Robotics and Automation, 34(1), 2019, doi:10.2316/J.2019.206-5422.[9] Z. Shams and S. Seyedtabaii, Nonlinear ﬂexible link robotjoint-fault estimation using TS-fuzzy observers, Interna-tional Journal of Robotics and Automation, 35(1), 2020, doi:10.2316/J.2020.206-0214.[10] Z. Geng, Study on the position control of electric cylinderbased on fuzzy adaptive PID, International Journal of Roboticsand Automation, 35(3), 2020, doi: 10.2316/J.2020.206-5226.[11] Q. Zhou, H.Y. Li, and P. Shi, Decentralized adaptive fuzzytracking control for robot ﬁnger dynamics, IEEE Transactionson Fuzzy Systems, 23(3), 2015, 501–510.[12] C. Ham, Z. Qu, and R. Johnson, Robust fuzzy control for robotmanipulators, IEE Proceedings-Control Theory Applications,147(2), 2000, 212–216.[13] Y.Q. Wei, J.D. Zhang, L. Hou, et al., Backstepping adaptivefuzzy control for two-link robot manipulators, IJCSI Inter-national Journal of Computer Science Issues, 10(2), 2013,303–308.[14] M.M. Azimi and H.R. Kooﬁgar, Adaptive fuzzy backsteppingcontroller design for uncertain underactuated robotic systems,Nonlinear Dynamics, 79(2), 2015, 1–12.7[15] Z.J. Yang and H.G. Zhang, A fuzzy adaptive tracking controlfor a class of uncertain strick-feedback nonlinear systems withdead-zone input, Neurocomputing, 272, 2018, 130–135.[16] D.A. Gabriel, Shoulder and elbow muscle activity in goal-directed arm movements, Experimental Brain Research, 116(2),1997, 359–366.[17] N.T. Antony and P.J. Keir, Eﬀects of posture, movement andhand load on shoulder muscle activity, Journal of Electromyo-graphy and Kinesiology, 20(2), 2010, 191–198.[18] T.L. Shen, Robust control fundamentals of robot (Beijing:Tsinghua University Press, 2000).[19] Y.X. Wu and C. Wang, Adaptive neural network control andleaning for uncertain robot, Control Theory & Applications,30(8), 2013, 990–997.[20] S. Malagari and B.J. Driessen, Globally exponential con-troller/observer for tracking in robots with DC motor dynamicsand only link position measurement, International Journal ofModelling, Identiﬁcation and Control, 19(1), 2013, 1–12.[21] V.M. Hernandez-Guzman and J. Orrante-Sakanassi, GlobalPID control of robot manipulators equipped with PMSMS,Asian Journal of Control, 20(1), 2018, 236–249.[22] Z.H. Man and M. Palaniswami, Robust tracking control forrigid robotic manipulators, IEEE Transactions on AutomaticControl, 39(1), 1994, 154–159.[23] K. Mu, C. Liu and J. Z. Peng, Robust tracking control forrobotic manipulator via fuzzy logic system and H∞ approaches,Journal of Control Science and Engineering, 2015, 2015, ArticleID 459431, 10 pages.[24] W. Ha and J. 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[33].3.4 Fuzzy ApproximatorsThis subsection will take the trouble to introduce the fuzzyapproximators as some references do to provide a trouble-free design process. It is very meaningful to assume anuncertain system as y = f(ω). ω ∈ Rnis an input vector,from which a mapping is performed to an output variabley ∈ R by employing the fuzzy IF-THEN rules in the fuzzyinference engine. The ith fuzzy rule is deﬁned asRi: If ω1 is μi1 and · · · and ωn is μin, then y is Biwhere μi1, . . . , μin and Biare fuzzy sets deﬁned by fuzzymembership functions. When the singleton fuzziﬁer, theproduct-inference engine and the centre-average defuzziﬁerare used, the variable y can be deﬁned asy =Ni=1 ˜yi nj=1 μij(ωj)Ni=1nj=1 μij(ωj)= θTψ(ω) (15)where N is the total number of fuzzy rules, θ =˜y1, . . . , ˜yN Tis an adjustable parameter vector with each˜yibeing the point at which the fuzzy membership function3shows the maximum, and ψ(ω) = [ψ1, . . . , ψN ]Tis a fuzzybasis vector whose lth element is of the following form:ψl =nj=1 μij(ωj)Ni=1nj=1 μij(ωj)(16)It is constructive to divide the uncertainty f (q, ˙q, ¨q)into an addition of two functions before applying therecommended fuzzy systems (15), thereby reducing thenumber of fuzzy rules. The addition equation is of the fol-lowing form:f(q, ˙q, ¨q) = f1(q, ˙q) + f2(q, ¨q) (17)where f1(q, ˙q) = −Δh(q, ˙q) and f2(q, ¨q) = −ΔM(q)¨q.Then these two functions can be approximated by theintroduced fuzzy systems (15), and their approximationsexpress, respectively, as follows:ˆf1(q, ˙q|θ1) = θ1T1 F1(q, ˙q), . . . , θ1Tn F1(q, ˙q)T= θ1TF1(q, ˙q)ˆf2(q, ¨q|θ2) = θ2T1 F2(q, ¨q), . . . , θ2Tn F2(q, ¨q)T= θ2TF2(q, ¨q)(18)The total approximation ˆf(q, ˙q, ¨q) is gotten after sum-ming ˆf1(q, ˙q|θ1) and ˆf2(q, ¨q|θ2). Here the minimum ap-proximation errors ε1and ε2also need to be given byε1= f1(q, ˙q) − θ1∗TF1(q, ˙q)ε2= f2(q, ¨q) − θ2∗TF2(q, ¨q)(19)where θ1∗and θ2∗are the optimal parameter matrices ofeach fuzzy logic system, respectively. ε1and ε2satisfythe following assumption according to the approximationprinciple.Assumption 4. For some known positive constants σ1and σ2, there areε1i ≤ σ1, ε2i ≤ σ2This subsection will be end after giving the followingapproximation errors of weight matrices:˜θ1= θ1− θ1∗, ˜θ2= θ2− θ2∗and their derivatives are˙˜θ1= ˙θ1and˙˜θ2= ˙θ2, respectively.3.5 Stability AnalysisFor designing a well-deﬁned adaptive fuzzy controller con-sidering the eﬀects of the uncertainties in DC motor, theapproximation errors and the external disturbances, thecompensation term u2 in (14) is developed asu2 = −λsgn(δ) (20)Deﬁne δ = [P21 P22] S as the ﬁltered errors, whereP21 ∈ Rnand P22 ∈ Rnare parts of the positive deﬁnitematrix P in (11), which is of the following form:P =⎡⎣P11 P21P21 P22⎤⎦ (21)λ is a gain matrix deﬁned by the following equation:λ = k−10 (σ1+ σ2+ u1 σg+ d0)In (22)and sgn(δ) ∈ Rnis a symbolic function vector made up bythe following n-scalar symbolic functions:sgni(δi) =⎧⎪⎪⎪⎨⎪⎪⎪⎩1 δi > 00 δi = 0−1 δi < 0,(23)Combining the actual manipulator system (6) with thecontroller (14) accompanied by (12) and (20)–(23), we getthe following ﬁnal error linear closed-loop state equation:˙S = AS + B (24)where B = [0TbT]Twith b = −˜θ1TF1(q, ˙q)−˜θ2TF2(q, ¨q)+εgu1 +Ku2 +ε1+ε2+τd, A and S are the same as in (10).Then, the stability of the closed-loop system (24) can bedescribed by the following theorem.Theorem. Considering Assumptions 1–4, the ﬁnallinear closed-loop state equation (24) obtained from(6), (12), (14) and (20)–(23) is asymptotically stable.Proof: Choose the weight matrix updating algo-rithms as˙θ1= η1F1(q, ˙q)δT˙θ2= η2F2(q, ¨q)δT(25)where η1 and η2 are positive constants.Then construct the following Lyapunov function:V =12STPS +12tr ˜θ1Tη−11˜θ1+12tr ˜θ2Tη−12˜θ2(26)By diﬀerentiating it with respect to time along theclosed-loop state (24), considering the aforementionedLyapunov equation (11) and its solution (21), plus thedeﬁnition δ = [P21 P22]S, we get the following:˙V = −12STQS − F1T(q, ˙q)˜θ1δ − F2T(q, ¨q)˜θ2δ+ εgu1 + Ku2 + ε1+ ε2+ τdTδ+ tr ˜θT1 η−11˙˜θ1 + tr ˜θT2 η−12˙˜θ2Then, considering (25) and using Lemma 2, we get thefollowing:˙V = −12STQS + εgu1 + Ku2 + ε1+ ε2+ τdTδ4Further, retrospecting (20)–(22) and aforementionedassumptions, ˙V becomes as follows:V = −12STQS + εgu1 − k−10 (σ1+ σ2+ u1 σg+ d0)Ksgn(δ)+ ε1+ ε2+ τdTδFurthermore, after using (23), ˙V becomes again asfollows:˙V = −12STQS + εgu1 − k−10 (σ1+ σ2+ u1 σg+ d0)[k1sgn1(δ1), . . . , knsgnn(δn)]T+ ε1+ ε2+ τdTδConsidering (12), for the ith element of the secondpart of ˙V , when δi ≥ 0, there isεgi u1i − kik−10 (σ1+ σ2+ u1 σg+ d0) + ε1i + ε2i + τdi ≤ 0and when δi ≤ 0, there is also the following:−(εgi u1i +kik−10 (σ1+σ2+ u1 σg+d0)+ε1i +ε2i +τdi) ≤ 0In other words, the second part of ˙V is never greaterthan zero. Then, there is always˙V ≤ −12STQS ≤ 0and limt→∞ S = 0. In fact, if we let V1(t) = V (t) −t0(V + (1/2)STQS)dτ, it is easy to prove ˙V1 = −1/2STQSuniform and continuous [30]. Using the Lemma 1, we canconclude limt→∞˙V1 = 0, whereupon S → 0 as t → ∞.So, the linear closed-loop error state equation (24) isasymptotically stable.The architecture of the proposed control scheme isshown in Fig. 1.Figure 1. Architecture of the proposed control scheme.Figure 2. Two-link robotic manipulator actuated by DCmotors.Table 1Nominal Values and Limits of Uncertaintiesm10 1 kg Δm1 0.1 kgm20 1 kg Δm2 0.1 kg4. Simulation ResultsTo verify the feasibility of the designed controller, a two-link manipulator is taken to give the simulation results.The conﬁguration of the manipulator actuated by DCmotors is shown in Fig. 2
[34],
[35].The manipulator model entries in (1) are described byM(q) =⎡⎣M11 M12M21 M22⎤⎦, h(q, ˙q) =⎡⎣h1h2⎤⎦, τd =⎡⎣0.5 cos(t)0.5 sin(t)⎤⎦where M11 = (m1 + m2)l21, M22 = m2l22, M12 = M21 =m2l1l2 cos(q1 − q2), h1 = m2l1l2 sin(q1 − q2) ˙q22 − (m1 +m2)l1 sin(q1)g, h2 = m2l1l2 sin(q1 − q2) ˙q21 − m2l2 sin(q2)g.Without the loss of generality, the modelling uncer-tainties are only embodied in m1 and m2. Then the nom-inal values of m1, m2 and their limits of uncertainties areshown in Table 1.5Figure 3. Simulation results when the errors are 0.1 kg: (a) position tracking of joints and (b) control input of joints.Figure 4. Simulation results when the errors are 0.3 kg: (a) position tracking of joints with K1 = 30I2, K2 = 5I2, (b) controlinput of joints with K1 = 30I2, K2 = 5I2, (c) position tracking of joints with K1 = 50I2, K2 = 10I2 and (d) control input ofjoints with K1 = 50I2, K2 = 10I2.The initial angles and their derivatives are selectedas q1(0) = q2(0) = 0, ˙q1(0) = ˙q2(0) = 0, respectively, the de-sired reference trajectories are chosen as qd1 = sin t andqd2 = 2 sin t. Another parameters are K1 = 30I2, K2 = 5I2,σ1= σ2= σg= 1, η1 = η2 = 0.1, d0 = 0.5, Q = I4, l1 = l2 = 1,k1 = k2 = 10.5, k0 = 10, g = 9.8.In the simulation, we adopt ﬁve fuzzy levels, i.e., NB,NS, ZO, PS, PB on the universe of each input variable anduse the following Gaussian membership function:μAli(xi) = exp −xi − ¯xliπ/242where ¯xli are −π/6, −π/12, 0, π/12, and π/6, respectively.The simulation results are shown in Fig. 3. Part (a) of Fig. 3is the trajectory tracking and part (b) is control input.When uncertainties increase, such as the upper limitsof quality errors here are 0.3 kg, but other parameters haveno changes, the simulation results are shown in Fig. 4(a)6Figure 5. Simulation results when σg= 5: (a) position tracking of joints and (b) control input of joints.and (b). From Fig. 4(a), we can see the tracking errorsincrease signiﬁcantly. However, if we enhance the gainmatrices K1 and K2, the tracking errors will reduce again.Parts (c) and (d) of Fig. 4 are simulation results withK1 = 50I2, K2 = 10I2.When the uncertainties of motor drive parameters in-crease, i.e., σgis bigger, but other parameters have nochanges, the tracking errors are aﬀected slightly. Never-theless, the volatility of control input will increase signiﬁ-cantly. Parts (a) and (b) of Fig. 5 are control input resultswhen the quality errors are 0.1 and 0.3 kg, respectively, un-der the condition of K1 = 30I2, K2 = 5I2 and σg= 5. Theseﬁgures tell us in the practical application, it is necessaryto get more accurate estimated values of the motor drivingparameters.5. ConclusionConsidering the DC motor drive system, an adaptive fuzzycontroller is designed for the tracking control problemof manipulator. First, the robotic manipulator systemcombines the electrical mathematical system to get themodel that is discussed in this article. Second, the feedbackcontrol technique is used to transform the model obtainedearlier into a concise state equation. Then, based onthe linear control method, the fuzzy control combinesthe adaptive sliding mode technique to get the proposedadaptive fuzzy controller. Last, the eﬀectiveness is notonly proved by using the Lyapunov stability theory butalso shown by taking a two-link manipulator as an exampleof simulation.Compared with most existing research results, thisarticle not only considers the modelling errors and externalinterferences of the manipulator system but also considersthe uncertainties of the motor drive system, thus makingour research results closer to the actual operating systemof the manipulator. Although the designed controller is tosolve the tracking problem of manipulator system in thisarticle, this design method is also suitable for many multi-input and multi-output systems which possess the similardynamic characteristics.References[1] K.J. Cheng and P.Y. Cheng, Investigation on velocity perfor-mance deviation of serial manipulators resulted from fabrica-tion errors, International Journal of Robotics and Automation,32(4), 2017, 324–332.[2] Z.X. Yang and D. 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