Improved Implementation of NonlinearAnalytical Redundancy Relations:

Application in Robotics ⋆

A. Villanueva ∗ C. Verde ∗ L. Torres ∗,∗∗

∗ Universidad Nacional Autonoma de Mexico, Coyoacan DF 04510,Mexico,

jvillanuevap@iingen.unam.mx,verde@unam.mx,ftorresO@iingen.unam.mx∗∗ Catedras CONACYT

Abstract:This paper copes with the problem of detecting actuators’ faults in a two-degree of freedomrobot manipulator by using nonlinear analytical redundancy relations composed by derivativesof known signals. To generate the residuals, the redundancy relations are handled with twoprocedures. The first one allows the determination of a transformation of the base relationsto reduce by one the order of the involved derivatives. The second one directly estimates thederivatives of the measurements by using a uniform robust exact differentiator. Simulationresults are presented to show the robustness of the nonlinear analytical redundancy relationsby using the transformation procedure even in the case of noisy measurements.

Keywords: Fault Diagnosis, Robot manipulator, Nolinear Analytical Redundancy Relations,Robust Differentiator, Actuator faults.

1. INTRODUCTION

The development of fault tolerant systems has been inthe past an extensive research field, mainly because of thesafety requirements of complex systems which include therobotics area.

The key for fault detection and identification (FDI) is thegeneration of signals, so-called residuals r(t), which mustsatisfy the following conditions:

• r(t) = 0 when the system operates in normal condi-tions, and

• r(t) = 0 in the presence of specific faults, at least inan interval of time.

In the linear models context, the most common toolsfor generating these residuals analytically are (i) stateobservers (Frank, 1994), (ii) identification algorithms (Is-ermann, 2010) and (iii) the calculation of analytic re-dundancy relations (ARR) based on analytical models(Gertler, 1991).

In the nonlinear models context, state observers havedominated the procedures for generating residuals withsliding modes (Edwards et al., 2000) and geometric tools(De Persis and Isidori, 2001). In some cases, however,the models of the systems do not satisfy the requiredconditions or do not have the suitable structure for thedesign of state observers. Moreover, the determination ofthe useful set of subsystems for diagnosis is not easy forlarge dimension systems.

The solvability of a FDI task for complex systems canbe analyzed by using the Redundancy Graph Concept⋆ Funded by: II-UNAM, IT100414-DGAPA-UNAM,CONACyT

(Verde et al., 2013). This framework has two advantages:allows the management of complex systems and is genericsince it does not require parameter information. Theimplementation of the residuals, however, is not a simpletask for a nonlinear system.

The derivation of the generic nonlinear analytical redun-dancy relations (NLARRs) is given in Blanke et al. (2006),which assumes that the state can be calculated from mea-surements by using the implicit function theorem. Theseexpressions depend in some cases on the derivatives ofsignals which can be inputs or outputs, faults and dis-turbances. To the best of our knowledge, the main reasonthat the NLARRs are not used to generate the residualsin a real application is the absence of robust algorithmsto compute the derivatives of the required signals and theabsence of a formal systematization for implementing theseNLARRs.

A procedure for obtaining NLARRs was proposed byHalder and Sarkar (2006), who considered models withan input affine structure, from which it is possible to findweighting matrices that make residuals more sensitive tofaults than perturbations; however they implement thederivatives with a differentiator which is sensitive to noisefrom the signals.

By assuming the existence of the NLARRs, Alwi andEdwards (2011) worked with an adaptable differentiatorbased on the work of Levant (1998) to generate the re-quired derivatives for the implementation of the NLARRs.

Fliess et al. (2004) used the flat systems’ properties todetect faults; the non-uniqueness property of the flatoutputs is employed for increasing the number of residuals.

Recently Frisk (2005) proposed reducing by one the orderof the derivatives on NLARRs by using transformations tosimplify their implementation and to generate a realizationin a state space; however the procedure is limited to thespecific structure of the NLARR.

The problems described above to generate robust residualsfor nonlinear systems by using analytical models motivatedthis work in which the implementation of the NLARRsobtained for a robot manipulator is discussed, when secondorder derivatives are involved. In particular the residualsare compared by using (i) a reduction of the derivativeorder in NLARRs by a state space realization and (ii)a robust differentiator without any transformation. Theresults show that the reduction of the derivative order inthe NLARR improves the residual behavior by achievinga high sensitivity factor.

The paper is organized as follows. In Preliminaries, thebackground for obtaining a residual generator by using theNLARR is introduced for the case of actuators’ faults. Thethird Section describes the model of the robot with andwithout faults, and the specific residuals for faults in theactuators of the manipulator are proposed. In the fourthSection, the residuals’ behavior is shown, and finally, someconclusions are drawn in Section 5.

2. PRELIMINARIES

2.1 Generation of NLARRs for Actuator Faults

By considering a multivariable input affine nonlinear dy-namic system of the form

x =f(x) + g(x)u,

y =Cx,(1)

where the state x is defined in an open set U of ℜn,u = [u1u2...uq] ∈ ℜq, y ∈ ℜm is the process output, and Cis a m× n output matrix. The functions f, g1, ..., gq ∈ ℜn

are valuated smooth mappings defined in the open subsetU and g(x) = [g1, g2, ..., gq]

′ ∈ ℜn×q.

In the presence of actuator faults, the input vector can berepresented by

u = un + fu, (2)

where un represents the fault-free input and fu representsthe fault vector; fu = 0 when the system is operating innormal conditions. The mathematical model (1) includingthe faults can be rewritten as

x =f(x) + g(x)un + g(x)fu

y =Cx.(3)

To design the NLARR in general the following assump-tions are considered.

• A1. The fault-free system (3) is asymptotically stableand could include the control loop.

• A2. The system (1) is observable.

The NLARRs are obtained with the following two-stepprocedure.

2.2 Step 1: Derivation of the Outputs

By considering model (3) and assuming that all the func-tions are differentiable with respect to their arguments, itis posible to construct the derivative of y

y = Cx. (4)

In addition, by replacing x by its expression, one gets

y = C(f(x) + g(x)un + fu). (5)

By iterating this process until some order of derivationq ≥ n

p − 1, one obtains the set of (q + 1)p equationsyy...

y(q)

=

CxCx...

Cx(q)

=

Cx

C(f(x) + g(x)un + g(x)fu)...

Cd(q)

dt(q)(f(x) + g(x)un + g(x)fu)

, (6)

which can be written as

y(q) = H(q)(x, u(q)n , f (q)

u ), (7)

where v(q) is a short notation for [v, v, ..., v(q)]′.

2.3 Step 2: Elimination of the State

By considering that system (3) is observable, one caneliminate the dependence of the state in (7) by determining

the value of q such that the Jacobian ∂H(q)

∂x has rank n.Thus to compute x, system (7) is decomposed into twosubsystems y(q∗)m

−−y(q∗)mn

=

H(q∗)m (x, un, fu)

−−−−−−−−H(q∗)

nm (x, un, fu)

, (8)

where the first subsystem of dimension n allows to com-

pute the state x as a function of un, y(q∗)m and fu by using

the implicit function theorem

x = ϕ(y(q∗)m , u(q)n , f (q)

u ). (9)

Moreover, the second subsystem is the rest of the deriva-tives which can be used to obtain the NLARRs.

Now by replacing (9) in the second subsystem of (8), withdimension (q + 1)p− n, one obtains an equivalent systemgiven by

y(q∗)nm = H(q∗)mn (ϕ(y(q∗)m , u(q)

n , f (q)u ), u(q)

n , f (q)u ), (10)

which is only a function of inputs, outputs and faultsignals.

If the system is operating in nominal conditions, fu is equalto zero, and (10) can be then written as

y(q∗)nm −H(q∗)nm (ϕ(y(q∗)m , u(q)

n , 0), u(q)n , 0) = 0. (11)

Thus, by defining the nominal case

H(qo)mn (ϕ(y

(q∗)m , u

(q)n , 0), u

(q)n ) := H

(q∗)mn (ϕ(y

(q∗)m , u

(q)n , 0), u

(q)n , 0) (12)

when fu is different from zero, one gets as well

y(q∗)nm −H(qo)mn (ϕ(y(q∗)m , u(q)

n , 0), u(q)n ) = 0, (13)

because

H(qo)nm (ϕ(y

(q∗)m , u

(q)n , 0), u

(q)n ) = H

(q∗)nm (ϕ(y

(q∗)m , u

(q)n , fu), u

(q)n , fu).

Thus, the relations (13) known as NLARRs can be usedto generate residuals, since they possess the followingqualities:

• Equal to zero if fu = 0.• Different from zero at least in a time interval if fu = 0.

Therefore, the set of residuals

r = y(q∗)nm −H(qo)mn (ϕ(y(q∗)m , u(q)

n ), u(q)n ) (14)

can be implemented to detect faults.

One can see from (14) that in some cases it could benecessary to know some derivatives of the known signalsfor their implementation. To manage this problem, in thiswork two methodologies are applied; an algorithm for dif-ferentiating based on sliding modes and a transformationof the NLARRs by using a proper filter and reducing theorder of the redundancy relation.

2.4 Uniform Robust Exact Differentiator

To generate derivatives of a signal, a differentiator basedon sliding modes is used (Cruz-Zavala et al., 2011) whereit is assumed that the signal to be differentiated is decom-posed as

s = s0 + ϑ, (15)

where s0 is the base signal, and ϑ is a uniformly boundednoise signal.The construction of the differentiator startsfrom the auxiliar model

z0 = z1

z1 = 0,

with z1 = s and a constant L such that |z| = |s| < L.Based on this model an observer is designed to estimatez1. The structure of the observer is given by

z0 = z1 − k1ϕ1(σ0) (16)

z1 =−k2ϕ2(σ0), (17)

where σo = zo−so represents the estimation error. By ap-plying the generalized super-twisting algorithm, one getsthe conditions k1 > 0 and k2 > 0 which are the observergains designed to guarantee the convergence of the dif-ferentiator. Furthermore the discontinuous functions aregiven by

ϕ1(σ0) = µ1 | σ0 |(1/2) sign(σ0) + µ2 | σ0 |(3/2) sign(σ0)

ϕ2(σ0) =µ21

2| σ0 | +2µ1µ2σ0 +

3

2µ22 | σ0 |2 sign(σ0)

with µ1 > 0, µ2 > 0, k1 ≤ 2√L and k2 >

k21

4 + 4L2

k21.

2.5 NLARR Realization

To implement the NLARRs, Frisk (2005) formally stud-ied the properties of the NLARRs, and he developed amethodology to reduce the number of derivatives to im-plement these relations.

Given a NLARR described by,

c(z, z) = 0. (18)

A realization in state space for the NLARR exists if itssatisfies the following conditions.

• (18) can be carried into the form:

c(z, z) =

n∑i=1

gi(z)zi + ν(z). (19)

• A potential exists for the vector fieldg(z) = (g1, ..., gn(z)), such that

dλ(z)

dt=

n∑i=1

gi(z)zi. (20)

The existence of the potencial λ is subject to the condition

∂gi∂zj

− ∂gj∂zi

= 0. (21)

If the above conditions are satisfied, one can rewrite (19)in the form:

c(z, z) =dλ(z)

dt+ ν(z). (22)

By filtering (22) through a nonlinear dynamic system onegets

r + h(r, z) =dλ(z)

dt+ ν(z), (23)

which can be written in a state space realization. Bychoosing a change of variables ω = r+λ and by regroupingterms in (23), one can write

ω = −h(r, z) + ν(z) (24)

r = ω − z, (25)

where for simplicity h(r, z) is chosen by a linear functionas βr.

3. CASE STUDY

By considering a robot of two-degree of freedom as inFig. 1, where the motion equation of the manipulator(Manrıquez, 2009) is given by

M(Θ)Θ +Co(Θ, Θ)Θ + g(Θ)+ fr(Θ, Θ) = ug + uf , (26)

where Θ = [θ1, θ2]T , M(Θ) ∈ ℜ2×2 is the inertia matrix,

Co(Θ, Θ) ∈ ℜ2×2 is the Coriolis and centrifugal matrix,

fr(Θ, Θ) ∈ ℜ2×1 is the vector associated with the friction,g(Θ) ∈ ℜ2×1 is the vector associated with the gravity,ug ∈ ℜ2×1 is the nominal torque vector, and uf ∈ ℜ2×1

represents the torque vector generated by faults in theactuators.

Fig. 1. Two-degree of freedom robot.

The vectors and matrices in (26) are given by

M(Θ) =

(M11(Θ) M12(Θ)M21(Θ) M22(Θ)

)M−1(Θ) =

(α(Θ) β(Θ)γ(Θ) δ(Θ)

)

Co(Θ, Θ) =

(C11(Θ, Θ) C12(Θ, Θ)

C21(Θ, Θ) C22(Θ, Θ)

)g(Θ) =

(g1(Θ)g2(Θ)

)

fr(Θ, Θ) =

(fr1(Θ, Θ)

fr2(Θ, Θ)

).

For simplicity the dependence of Θ and Θ in the matricesM−1(Θ), Co(Θ, Θ), fr(Θ, Θ), g(Θ) are no longer writtenexplicitly.

The assumed tracking signal for the robot is a circulartrajectory in the x-y plane, as revealed in Fig. 2.

Fig. 2. Trajectory of the manipulator robot.

In this work the faults considered generate only a partof the desired torque; moreover, these kinds of faults aregenerated by the degradations of the actuators that modifythe trajectory of the robot. This faulty scenario is shownin Fig. 3.

The aim is to detect faults with the behavior shown in Fig.3; to achieve that it is necessary follow the steps given inthe previous section.

By defining the state x = [θ1 θ2 θ1 θ2]T , the model (26)

can be rewritten as

Fig. 3. Trajectory of the manipulator robot with faults.

x1 = x3

x2 = x4

x3 = αug1 + βug

2 − (αC11 + βC21)x3− (αC12 + βC22)x4

−αg1 − βg2 − αfr1 − βfr2 + αuf1 + βuf

2

x4 = γug1 + δug

2 − (γC11 + δC21)x3− (γC12 + δC22)x4

−γg1 − δg2 − γfr1 − δfr2 + γuf1 + δuf

2 .

By assuming that the positions x1 and x2 are measurable,the output becomes

y =

(1 0 0 00 1 0 0

)x. (27)

Thus, to construct the NLARRs, one computes the deriva-tive of the signal y two times, and one attains the statemodel

y1 = x1

y2 = x2

y1 = x3 (28)

y2 = x4

y1 = αug1 + βug

2 − (αC11 + βC21)x3− (αC12 + βC22)x4

−αg1 − βg2 − αfr1 − βfr2 + αuf1 + βuf

2

y2 = γug1 + δug

2 − (γC11 + δC21)x3− (γC12 + δC22)x4

−γg1 − δg2 − γfr1 − δfr2 + γuf1 + δuf

2

of dimension six, with its Jacobian matrix of rank four.Therefore, only two derivatives are required to eliminatethe state x in (28).

As the second step, from the first four equations of (28)the state can be evaluated, since

x1 = y1

x2 = y2 (29)

x3 = y1

x4 = y2.

Finally by substituting the vector (29) in the last twoequations of the system (28), one gets

y1 = (αC11 + βC21)y1 − (αC12 + βC22)y2 − αg1 − βg2

− αfr1 − βfr2 + αug1 + βug

2 + αuf1 + βuf

2 (30)

y2 = (γC11 + δC21)y1 − (γC12 + C22)y2 − γg1 − δg2

− γfr1 − δfr2 + γug1 + δug

2 + γuf1 + δuf

2 . (31)

One can see that in both equations the terms uf1 and

uf2 appear, which are associated with the actuator faults;

therefore, one can define as base residuals

r1 = y1 + (αC11 + βC21)y1 + (αC12 + βC22)y2

+ αg1 + βg2 + αfr1 + βfr2 + αug1 + βug

2 (32)

r2 = y2 + (γC11 + δC21)y1 + (γC12 + C22)y2

+ γg1 + δg2 + γfr1 + δfr2 + γug1 + δug

2, (33)

which depend on the second derivative of the outputvector. This means the residuals take the form

r1 = c1(y1, y1, y2, y1, y2) (34)

r2 = c2(y2, y1, y2, y1, y2). (35)

3.1 Implementation by a NLARR’s Realization

To apply the procedure given in Section 2.5, first onetransforms (34) and (35) by

z =

∫y dt

z =

∫y dt

A =

∫y dt,

thus obtaining the transformed residuals

r1t = z1 + (αC11 + βC21)z1 + (αC12 + βC22)z2

+ αg1 + βg2 + αfr1 + βfr2 + αug1 + βug

2 (36)

r2t = z2 + (γC11 + δC21)z1 + (γC12 + C22)z2

+ γg1 + δg2 + γfr1 + δfr2 + γug1 + δug

2 (37)

which clearly have the form required for the state spacerealization:

c1t(z1, z1, z2) = 0 (38)

c2t(z2, z1, z2) = 0. (39)

Therefore, by filtering these residuals

rκ + βrκ = z1 + υ(z) (40)

rξ + βrξ = z2 + υ(z), (41)

one can realize (40) and (41) as

ω1 =− βω1 − βz1 + υ(z)

rκ =ω1 − z1(42)

ω2 =− βω2 − βz2 + υ(z)

rξ =ω2 − z2(43)

holding the outputs rκ and rξ the properties of the originalresiduals.

3.2 Implementation by the Exact Differentiator

To implement (32) and (33), the derivatives of y1, y2, y1, y2are replaced by their estimation by using a uniform robustexact differentiator, which obtains

r1b = ¨y1 − (αC11 + βC21) ˙y1 − (αC12 + βC22) ˙y2

− αg1 − βg2 − αfr1 − βfr2 + αu1 + βu2 (44)

r2b = ¨y2 − (γC11 + δC21) ˙y1 − (γC12 + C22)y2

− γg1 − δg2 − γfr1 − δfr2 + γu1 + δu2 (45)

where ˙y1 is calculated according to the algorithm

z0 =z1 − k1ϕ1(z0 − y1)

z1 =− k2ϕ2(z0 − y1)

˙y1 =z1.

Furthermore, ˙y2 is given by

z02 =z12 − k1ϕ1(z02 − y2)

z12 =− k2ϕ2(z02 − y2)

˙y2 =z12.

Note that the second derivatives of y1 and y2 are calculatedby using the differentiator two times.

4. RESULTS

In this section the results of implementing the residualsobtained for the case study are shown. The results were

simulated by considering that the faults uf1 and uf

2 appearat 100[s]; the magnitude of the faults corresponds to adeviation of 75% of the nominal torque. In addition, whitenoise is assumed in the measurements.

The responses of the three residuals (32), (42) and (44),

by considering the fault uf1 , are shown in Fig. 4. Note

that the base residual (32) has been as well implementedby the Simulink R⃝ differentiator as the reference pattern(MATLAB R2008, 2008).

The implementation of the residuals (33), (43) and (45),

by considering the fault uf2 , are shown in Fig. 5, where (33)

has been implemented by the Simulink R⃝ differentiator.

One can see from Figs. 4 and 5 that by using exact differ-entiators the quality of the residual improves as comparedwith a simple differentiator of Simulink R⃝; however, anadvantage by using a state space realization is that theresiduals are more sensitive than the robust differentiator.The noise residuals obtained with the differentiator ofSimulink R⃝ have a poor performance from the FDI pointof view.

5. CONCLUSIONS

A fault detection system for a two-degree of freedom robotwas presented. When nonlinear analytical redundancy

Fig. 4. Residuals when the fault uf1 appears.

Fig. 5. Residuals when the fault uf2 appears.

relations are obtained these relations can be functions ofderivatives of known signals. To solve this problem, twoprocedures were used. The results show the advantage ofthe realization in state space. By reducing the order ofthe derivatives the results are more sensitive than thoseobtained by using differentiators.

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