# Development of Fault Identification System for Electric Servo Actuators of Multilink Manipulators Using Logic-Dynamic Approach

Authors: V. Filaretov; A. Zuev; A. Zhirabok; A. Protsenko
Journal: Journal of Control Science and Engineering
 (2017)

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## Abstract

This paper presents the method of synthesis of faults identification systems for electric servo actuators of multilink manipulators. These actuators are described by nonlinear equations with significantly changing coefficients. The proposed method is based on logic-dynamic approach for design of diagnostic observers for fault detection and isolation. An advantage of this approach is that it allows studying systems with nonsmooth nonlinearities by linear methods only. For solving the task of faults identification, a residual signal feedback was proposed to be used for observers. The efficiency of the proposed fault identification system was confirmed by results of simulation.


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## Body

## 1. Introduction

At present, the use of multilink manipulators (MM) has considerably extended and the operations which are carried out by them become more complicated [1–4]. The occurrence of any faults in actuators of MM may lead to significant aggravation of quality of performance of the MM operations and reduction of accuracy. Therefore, the reliability and safety of such complex equipment at operating in autonomous mode have the significant importance. Today the most typical actuators of MM are electric servo actuators. One way to improve the reliability of electric servo actuators of MM is using real-time fault diagnosis and fault tolerant control [1]. Fault tolerant control is based on forming special control, providing automatic stabilization of the main parameters of actuators when faults occur [2]. For providing such control, all possible faults should be timely detected and their values should be estimated.The problem of fault detection and isolation (FDI) was extensively investigated for the past 20 years; see, for example, the surveys [2, 5–8] and the books [9, 10]. Many problems have been studied and solved: different methods of residual generation and relationships including robustness and adaptive threshold test, H∞-approach, and fuzzy logic; many types of systems have been considered: linear, descriptor, and different classes of nonlinear systems. Many practical examples were considered; in particular, special book was devoted to robotic systems [11].As a rule, the electric servo actuators of MM are nonlinear essentially with such types of no differentiable nonlinearities as saturation, Coulomb friction, backlash, and hysteresis [12]. Therefore, one has to use nonlinear methods of the FDI. However, most of papers dealing with the FDI problem consider the nonlinear systems with differential nonlinearities [13–16]; therefore they cannot be used in our case.At present, there are several approaches for designing diagnostic observers allowing fault diagnosis for systems with no differentiable nonlinearities. Among them there are methods using the algebra of functions [17], the logic-dynamic approach (LDA) [18–20], and others [21–24]. Whereas the algebra of functions demands complex analytical calculations using symbolic mathematical packages, the LDA allows using the methods of linear algebra for nonlinear systems and provides relatively simple procedure for synthesis of diagnostic observers. The LDA includes three steps: replacing the original nonlinear system with appropriate linear system, constructing the set of linear diagnostic observers, and adding to these observes the nonlinear parts. But LDA can solve only FDI problem without fault identification. To estimate the value of the faults, one uses in [19, 20] a feedback in diagnostic observers. These methods provide the faults identification for only first-order diagnostic observers. But at case when electric servo actuators of MM have partially measured state vector, then there are no guarantees that observers will have first order.The present paper suggests a solution to the problem of developing the method of synthesis of fault identification (FI) system for electric servo actuators of MM which are described by mathematical model including differential equations with nonlinearities and partially measured state vector. The proposed method assumes the use of the diagnostic observers based on LDA with special feedback signal dependent on the residual for real-time identification of fault value.The paper is organized as follows. Section2 describes the model of electric servo actuators of MM. Section 3 is devoted to the synthesis of diagnostic observers for FDI using logic-dynamic approach. Firstly, the main steps of this approach are given and then the observers are described in detail. In Section 4, the methods of synthesis of FI system for created observers are presented. First, the analysis of mismatch of states of electric servo actuator and the observer is given and then the synthesis of residual signal feedback is shown. An example is considered in Section 5. Section 6 concludes the paper.


## 2. Description of Dynamics of Electric Servo Actuators of MM

In this paper, the MM with PUMA type kinematic scheme with 6 degrees of freedom is considered (see Figure1). Today, the MM with such kinematic scheme are mostly used in industry. Each degree has an electric servo actuator that includes DC motor, reducing gear and sensors measuring the electric current, angular speed of DC motor rotor, and angle of rotation of output shaft of reducing gear.Figure 1
MM with PUMA type kinematic scheme.It is expedient to use the second form Lagrange equation or other methods to obtain the moment torquesMi in each degree of freedom of the MM (see Figure 1) while moving its gripper along the complex spatial trajectory, if laws of change of generalized coordinates qi (i=1,6¯) of the MM are known.The expressions for moments acting in each degree of freedom of MM can be represented in the form [3](1)Mi=Hiiqq¨i+hiq,q˙q˙i+MEiq,q˙,q¨,where Hii(q) is the diagonal element of matrix of inertia of MM; hi(q,q˙) is the element of the vector of Coriolis and centrifugal forces; and MEi(q,q˙,q¨) is the torque which represents interactions acting toith degree of freedom from other MM’s links and also action of the gravitational forces. These values with the help of second Lagrange equation are defined. Further for simplicity we will not write the indexi.It is assumed that the following typical faults are possible in each electric servo actuator of the MM:(i)
The faultd1(t) caused by change of value of Coulomb friction moment M~сfg0 in reducing gear(ii)
The faultd2(t) caused by change of value of Coulomb friction moment M~сfm0 in DC motor(iii)
The faultd3(t) caused by change of value of electrical resistance R~.Considering (1) and these faults, model of electric servo actuator of the MM can be presented by the following differential equations: (2)x˙1=x2,x˙2=-cig2Hqx1-kvg+hq,q˙Hqx2-MEq,q˙,q¨Hq-Mcfg0Hqsign⁡x2+d1t,x˙3=x4,x˙4=cigJx1-cJx3-kvmJx4+kmJx5-Mcfm0Jsign⁡x4+d2t,x˙5=-kωLx4-RLx5+kuLu+d3t,where x1=a˙g and x2=ag are speed and angle of output shaft of reducing gear; x3=a˙m and x4=am are speed and angle of DC motor shaft; R, L, and x5=I are resistance, inductance, and current of electric motor rotor circuits, accordingly; kω is a coefficient of counter-EMF; km is a coefficient of motor moment;J is a rotational inertia of rotating parts of reduction drive and DC motor; c is rigidity of mechanical transmission; ig is a coefficient of the reducing gear; u is control signal; ku is an amplifier gain; Mсfm0 and Mсfg0 are coefficients of Coulomb friction of DC motor and reducing gear; and the variables dj (j=1,2,3) represent the effects of faults on the system and are of the form(3)d1t=-M~сfg0Hqsign⁡x2,d2t=-M~сfm0Hqsign⁡x2,d3t=-R~LI.Equation (2) has nonlinearities and significantly changing coefficients because of changes in parameters H(q), h(q,q˙), and ME(q,q˙,q¨).


## 3. Synthesis of Diagnostic Observers for FDI Using LDI

System (2) can be represented by the differential equation in a matrix form as follows: (4)x˙t=Fxt+Gut+Bx+Edt,yt=Cxt,where B(x) is a vector corresponding to the system nonlinearities, x∈R5, y∈R3, C∈R3×5, F∈R5×5, G∈R5, (5)F=01000-cig2Hq-kvg+hq,q˙HqcigHq0000010cigJ0-cJ-kvmJkmJ000-kωL-RL,E=000100000010001,C=100000001000001,B=0-MEq,q˙,q¨-Mcfg0sign⁡x2Hq0-Mcfm0sign⁡x4J0,x=aga˙gama˙mI,G=0000kuL.If faults occur during the work of MM, they should be promptly detected and identified. Then the fault tolerant control can be used. The traditional methods of diagnosis that focus on systems described by linear models cannot provide the solution to the problem of synthesis of FI systems for electric servo actuators of the MM described by model with the parametersH(q), h(q,q˙), and ME(q,q˙,q¨) that are significantly variable and under the presence of the nonlinear component B(x).In this paper, another synthesis method of FI systems is proposed. This method includes two basic steps [25]: (1) using LDA to construct the diagnostic observer [17, 18] which can detect occurrence of specific faults; (2) introducing the specific feedback in observers for identification of the fault values [19]. Below these stages are considered in more detail.The LDA to solve the FDI problem for system (4) includes the following steps [18]: (1) replacing the initial nonlinear system (4) by linear system; (2) solving the FDI problem for the linear system with some additional restrictions and obtaining the bank of linear observers; (3) transforming the linear observers into the nonlinear ones. Thus, the nonlinear diagnostic observers detecting the faults are of the following general form:(6)x˙∗t=F∗x∗t+G∗ut+B∗x∗+M∗yt,y∗t=C∗x∗t,where x∗(t)∈Rk is the state vector of the observer, y∗t is the output vector, F∗∈Rk×k, G∗∈Rk, B∗∈Rk, M∗∈Rk×3, and C∗∈Rk are constant matrices, andk is dimension of the observer; it is assumed that the matrices F∗ and C∗ can be represented in a canonical form (7)F∗=010⋯0001⋯0···000⋯0,C∗=100⋯0.The observer (6) generates the residual r(t)=Ny(t)-y∗(t) for certain row matrix N∈R3 which has to be determined. If there are no faults and d(t)=0, then r(t)=0; if a fault occurs, r(t)≠0. It is well-known from the linear FDI theory [2, 9, 26] that, for the observer design, the matrix Φ where Φx(t)=x∗(t) in the unfaulty case after the response to unlike conditions has died out plays the main role. In the absence of faults, the following well-known set of equations holds [2, 9, 26]:(8)C∗Φ=NC,ΦF=F∗Φ+M∗C,G∗=ΦG,B∗x∗=ΦBx.For each faultdi, the diagnostic observer Oi should be constructed. Matrices and vectors Φ, F∗, B∗(x∗), G∗, N, and M∗ describing the observer can be found in (8) [17]. As a result, the observers O1, O2, and O3 for the faultsd1,d2, andd3 were obtained:O1:(9)Φ=10000kvg+hq,q˙Hq100000cigHq00,M∗=-kvg+hq,q˙Hq00-cig2Hq000cigHq0,N=100,F∗=010001000,G∗=000,B∗=0-MEq,q˙,q¨+Mcfg0sign⁡x2Hq0.O2:(10)Φ=0001000-cJ00,M∗=cigJ-kvmJkmJ0-cJ0,N=010,F∗=0100,G∗=00,B∗=-Mcfm0sign⁡x4J0.O3:(11)Φ=00001,M∗=0-kωL-RL,N=001,F∗=0,G∗=kuL,B∗=0.As a result, the synthesized observersO1, O2, and O3 provide solving the FDI problem for electric servo actuators of MM. Clearly, the observers O1 and O2 have order more than one. This is caused by the fact that the electric servo actuator (2) has the partially measured state vector. Therefore, the known methods [19, 20] for FI cannot be used. Below the new synthesis method of FI systems for observers constructed by using LDA is proposed.


## 4. Synthesis of FI System for Observers Created by Using LDA

To solve the problem of FI for synthesized observers, consider how residualr(t) changes when the faults dj(t) occur. After differentiating the equation for residual and taking into account (4), (6), and (8), the following can be obtained:(12)r˙t=C∗F∗Φxt-x∗t+ΦBx-B∗x∗+ΦEdt.The vectore(t)∈Rk is introduced which is a mismatch of states of electric servo actuator and the observer:(13)et=Φxt-x∗t.Taking into account the vectorC∗, the residual r(t) is a first element of mismatch vector e(t):r(t)=e1(t). Therefore, (12) can be rewritten as (14)e˙t=F∗et+ΦBx-B∗x∗+ΦEdt.Analysis of (14) shows that if order of the obtained observer is one (i.e., y∗(t)=x∗(t)), then if faults occur, the following conditions are true: (1)F∗=0,  (2)e(t)=r(t)∈R1, and (3)B∗=ΦB. In this case, the feedback is proportional to the residual signal asTr(t) (T is the feedback coefficient) and allows identifying the faults value [19, 20]. However, in case when order of the observer is more than one, B∗≠ΦB due to x∗(t)≠Φx(t). Moreover, in (6) the components F∗e(t) and ΦB(x(t)) are unknown. As a result, it is impossible to make fault identification. To solve this problem, one proposes a feedback of residual signal which provides e(t)=0. In this case the value of this feedback is proportional to the value of fault. This provides a possibility of identifying the fault value.After the introduction of proposed residual signal feedback, the model of observer will be(15)x˙∗t=F∗x∗t+G∗ut+B∗x∗+M∗yt+wr,(16)y∗t=C∗x∗t,where w(r)∈Rk is the vector specifying feedback dependent on the residual signal.Considering the feedback (15), equation for the mismatch vector is as follows:(17)e˙t=F∗et+ΦBx-B∗x∗+ΦEdt-wr.Equation (17) can be rewritten in the form (18)e˙1t=e2t-w1r,e˙2t=e3t-w2r1,e˙3t=e4t-w3r,⋮e˙vt=ev+1t+fvdjt+av-wvr.⋮e˙kt=-wkr,where av is thevth element of the vector ΦB(x(t))-B∗(x∗(t)), v=1,k¯, and fv is thevth element of vector ΦL.Whenwv+1(r)=0,…,wk(r)=0 and the initial conditions of the electric servo actuator and the observer are agreed upon, ev+1(t),…,ek(t) are zero regardless of occurrence of the fault. Taking this into account, it is possible to rewrite (18) in the form of a differential equation:(19)fvdjt+av=e˙vt+wvr=e¨v-1t+w˙v-1r+wvr=e⃛v-2t+w¨v-2r+w˙v-1r+wvr=e1vt+w1v-1r+⋯+w1v-1r+wvr.Making elements of the feedback vector, except thevth element, proportional to the residual wn(r)=(Tn/T0)r(t) (n=1,v-1¯) and wv(r)=(Tv/T0)r(t)+(1/T0)z(r) (T0,…,Tv are feedback coefficients), one obtains(20)fvdjt+av=rvt+T1T0rv-1t+⋯+Tv-1T0r1t+TvT0rt+1T0zr.Feedback coefficients T0,…,Tv are chosen to guarantee the stability of observers [25].Considerfvdj(t)+av as input signal and r(t) as output; then (20) can be presented by a transfer function W=T0/(T0sv+⋯+Tv+zs), wheres is the symbol of Laplace transformation. Suppose that input fvdj(t)+av can be presented by a polynomial of degree q:fvdj(t)+av=b0+b1t+⋯+bqtq. The Laplace transformation of such input is Q=b0/s+b1/s2+⋯+bq/sq+1. Then the steady value of r(t) can be found as r(t)=lims→0=lims→0⁡WQs=T0(b0sq-1+⋯+bq)s/(T0sv+⋯+Tvsq+1+zssq+1). Assuming z(s)=Tv+1/s+Tv+2/s2+⋯+Tv+q/sq+Tv+q+1/sq+1, it is possible to obtain r(t)=lims→0⁡(T0(b0sq-1+⋯+bq)s/((T0sv+⋯+Tv)sq+1+Tv+1sq+⋯+Tv+q+1))=0.Thus, the introduction of such a feedback results in a steady value ofr(t)=0. This also results in resetting all elements of the vector e(t). For many real systems, value of the fault is constant or slowly changes and it is sufficient to introduce one integral of residual in feedback to achieve the required accuracy of the faults identification: z(r)=∫r(t)dt. Thus, after using this kind of feedback, residual tends to zero, after the completion of the transition process, including the event when the faults occur. This ensures the synchronization of state vectors of the electric servo actuator and the observer. Thus, faultdj can be found as(21)djt=1fvT0∫rtdt.After applying the proposed procedure for observersO1, O2, and O3, we receive the following feedback for fault identification:O1:(22)w=w11w120T,w11=T1T01rt,w12=T2T01rt+∫1T01rtdt.O2:(23)w=w120T,w12=T3T02rt+∫1T02rtdt.O3:(24)wr=T4rt.Finally, considering the expression (21), it is possible to identify the faults value for the electric servo actuators of the MM:(25)M~cfg0=-Hqsign⁡a˙gfvT0∫rtdt,M~cfm0=-Jsign⁡a˙mfvT0∫rtdt,R~=-rtLT4I.Thus, the problem of identification of value of faults has been solved.


## 5. Simulation of Performance of FI System

Analysis of effectiveness of developed FI system was implemented using PUMA type of the MM. The FI system was synthesized for electric servo actuator of second degree of freedom of MM (see Figure1). The results for other degrees of freedom of the MM are similar.The second degree of freedom of MM has the termsH, h, and ME as follows:(26)H=Jn2+Jn3+m2l2∗2+m3l3∗2+m3+mgl22+mgl32+2l2m3l3∗+mgl3cos⁡q3,h=-2l2m3l3∗+mgl3q˙3sin⁡q3,ME=Jn3+m3l3∗2+mgl32+m3l3∗+mgl3l2cos⁡q3q¨3-l2m3l3∗+mgl3q˙32sin⁡q3+gm2l2∗+m3+mgl2sin⁡q2+gm3l3∗+mgl3sin⁡q2+q3-12Jn2-Js2+m2l2∗2+m3+mgl22sin⁡2q2+Jn3-Js3+m3l3∗2+mgl32sin⁡2q2+q3+2l2m3l3∗+mgl3sin⁡2q2+q3q˙12,where Jsi and Jni are the moments of inertia of linki along its longitudinal and transverse axis passing through its center of mass, accordingly; mi is the mass of linki; mg is the mass of the load; li is the length of linki; li∗ is the length fromith joint to the center of mass ofith link; g is the acceleration of gravity; and qi, q˙i, q¨i are the angle of rotation of the linki and its derivatives.The following set of parameters of MM was used for simulation:l1=l2=l3=0.5 m, l1∗=0.1 m; l2∗=l3∗=0.2 m, mg=5 kg; m1=25 kg, m2=m3=5 kg; Js2=0.007 kg·m2, Js3=0.005 kg·m2, Jn2=0.14 kg·m2, Jn3=0.2 kg·m2, R=0.5 Ω, L=0.0005 H, kω=0.04 V·s, km=0.04 N·m/A, J=10-3 kg·m2, ig=100; ku = 800, Mcfg0=Mcfm0=0.02N·m, and kvg=kvm=0.005N·m·s/rad.The input signalsq1=sin⁡(0.2t), q2=u=sin⁡(0.5t), and q3=sin⁡(t) were applied to the electric servo actuator. The parameters of feedback for observers were chosen as follows: (27)T01=1.372×10-6,T02=2.564×10-4,T1=3.704×10-4,T2=0.033,T3=0.033,T4=100.Faults were presented by changing the following values:(i)
Coulomb friction coefficient in reducing gear by 50% (M~cfg0=0.02 N·m and M~cfg0=0.02sin⁡(0.5t) N·m) at the time from t1=5 s to t3=15 s(ii)
Coulomb friction coefficient in DC motor by 50% (M~cfm0=0.02 N·m and M~cfm0=0.02sin⁡(0.5t) N·m) at the time from t2=10 s to t4=20 s(iii)
Electrical resistanceR~ by 50% (R~=0.2 Ω and R~=0.2sin⁡(0.5t) Ω) at the time from t3=15 s to t5=25 s.Values of introduced faults (curve 1) and estimated value of fault (curve 2) are shown in Figures2–7. It follows from these figures that the constructed FI system provides accurate estimation of occurred fault. The results of simulation confirm the efficiency of the proposed method. Realization of the synthesized FI systems does not present any difficulties. These FI systems can be realized with the help of serial microprocessors.Figure 2
Estimation of constant faultM~cfg0 value by proposed FI system.Figure 3
Estimation of variable faultM~cfg0 value by proposed FI system.Figure 4
Estimation of constant faultM~cfm0 value by proposed FI system.Figure 5
Estimation of variable faultM~cfm0 value by proposed FI system.Figure 6
Estimation of constant faultR~ value by proposed FI system.Figure 7
Estimation of variable faultR~ value by proposed FI system.


## 6. Conclusion

In this paper, the method of synthesis of accurate FI systems for electric servo actuators of the MM, which are described by nonlinear differential equations with variable parameters and partially measured state vector, is suggested. This method consists in applying the logic-dynamic approach for synthesis of diagnostic observers guaranteeing detection and localization of possible faults and introducing of special feedback in diagnostic observers, allowing identifying values of faults. Efficiency of the proposed method was confirmed by the results of simulation. The subject of further research is application of the proposed method for synthesis of the fault tolerant control systems for actuators of the MM.



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