real-time robust adaptive control of robots subjected to actuator voltage constraint

Nonlinear DynDOI 10.1007/s11071-014-1574-z

ORIGINAL PAPER

Real-time robust adaptive control of robots subjectedto actuator voltage constraint

Alireza Izadbakhsh · Mohammad Mehdi Fateh

Received: 31 October 2012 / Accepted: 2 July 2014© Springer Science+Business Media Dordrecht 2014

Abstract This paper is concerned with the problemof design and implementation of a robust adaptivecontrol strategy for electrically driven robots whileconsidering to the constraints on the actuator volt-age input. The proposed approach provides a flexi-ble design framework and stable to deal with robust-ness compared with many other adaptive controllers,such as halting/slowing adaption techniques and adap-tively adjusting command signal, which are proposedfor robotic applications. The control design procedureis based on a new form of universal approximation the-ory and using Stone–Weierstrass theorem, to avoid sat-uration besides being robust against both structured andunstructured uncertainties associated with external dis-turbances and actuated manipulator dynamics. More-over, the proposed approach eliminates problems aris-ing from classic adaptive feedforward control scheme.The analytical studies as well as experimental resultsproduced using MATLAB/SIMULINK external modecontrol on a two degree of freedom electrically drivenrobot demonstrate high performance of the proposedcontrol schemes.

A. Izadbakhsh (B)Department of Electrical Engineering, College ofEngineering, Garmsar branch, Islamic Azad University,Garmsar, Irane-mail: [email protected]

M. M. FatehDepartment of Electrical and Robotic Engineering,Shahrood University of Technology, Shahrood, Iran

Keywords Robust adaptive control · Real-timecontrol · Actuator saturation · Model-free control

1 Introduction

The actuator input constraints are one of the majorproblems that arise while controlling an actuateddynamic system. These constraints are due to eitherphysical limitations of the devices, or practical reasons(security/safety) that restrict the command signal com-ing from controller to the actuators. The main challengeusually occurs when a large control signal is requiredfor obtaining good tracking performance of hard refer-ence trajectories. It can cause saturation of the actua-tor which can excite windup phenomenon, especiallywhen the controller includes the integral term in itsstructure [1]. One solution to this problem is utiliz-ing small control gains, but it usually leads to unsatis-factory tracking performance and robustness. Anotherapproach is dividing the reference trajectory into manysmall partitions called “profile move” [2]. However, itmay cause undesirable effects such as excessive over-shoots and long settling time. Moreover, it would not beable to utilize the full power of the driver. To deal withthese problems, many valuable torque-based controlstrategies have been proposed by researchers, aimingto prevent instability and nominal performance degra-dations of the robotic systems considering to input con-straints [3–11]. The considerable point is that, althoughthese approaches are satisfactory in principle, they areoften criticized for few reasons:

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A. Izadbakhsh, M. M. Fateh

– Almost, all of aforementioned control strategiesrequire extensive computations, as well as a prioriknowledge about the complex dynamics of robotswith a considerable accuracy, which are impossibledue to unknown (or approximated known) parame-ters of the system.

– In practice, a torque-based control law cannot begiven directly to the torque inputs of an electricalmanipulator. The torque control command cannotperfectly provided by the electrical motors [12,13].

– Almost, all of aforementioned control strategies donot consider the role of actuator dynamics in the con-trol design procedure. The fact is that actuators areoften a source of uncertainties, due e.g., to calibra-tion errors, or parameter variation from overheat-ing and changes in environment temperature [14].Ignoring the actuator dynamics, although, simpli-fies the control design problem; however, it involvesneglecting nonlinearities such as saturation, deadzone, backlash, and hysteresis which degrades theprior-designed control characteristic.

To tackle these problems, some related works in thefield of adaptive control have been proposed [15–20]. Moreover, several approaches for minimizing theperformance loss due to input constraints have beenreported (see [21] and reference there in). The firstapproach adaptively adjusts the command or feedbacksignal to avoid saturation. The second method is slow-ing down or suspending the adaptation process over thesaturation time. Because, continued adaptation duringinput saturation can lead to instability due to nonlin-ear and time-varying features of adaptation mechanism[22]. Finally, the third and more common approach isaugmenting of the error signal by a signal derived fromthe actual plant Input. However, there are yet problemsassociated with real-time adaptive feedforward controlcombined with saturation nonlinearity for the trajecto-ries and payloads that require large control gains forcontrol implementation:

– Computation requirements for real-time parameteridentification, and sensitivity to numerical accuracyand the existing noise increase in an undesirableform as the state variables increase.

– For computer controlled systems, the discretiza-tion process by zero-order holds usually leads to adiscrete-time non-minimum phase system with atleast one unstable zero [23]. Thus, stability consid-erations prohibit direct applications of self-tuning

and model reference adaptive control (MRAC) algo-rithms, which it also leads to pole-zero cancelations[24].

– Utilizing inverse model of the system for Feedfor-ward branch is not possible because of non analyticalstructure of the saturation function.

– Unstable zeros in the reference model cause chal-lenges such as significant phase errors over a broadrange of frequencies, due to problem arises fromFeedforward path implementation, which are notrealizable for saturated actuated manipulator sys-tems [24].

This paper presents a robust adaptive compensator,applicable for real motion control systems. Com-pared to other adaptive control strategies, the proposedapproach is free from actuated manipulator dynamic,considered here as un-modeled dynamic. It also consid-ers the external disturbances effects and time-varyingparameters, which is the main concern in conventionalMRAC [25]. It is discussed that these uncertainties canbe approximated by a simple p order linear differentialequation. Thus, it can be handled by means of simplewell-known MRAC techniques which greatly facilitatethe analysis and design task as well. As a result, theupdate laws do not need any parameters of the sys-tem. This paper is organized as follows. The model andproblem statement are presented in Sect. 2. Section 3 isdevoted to the description of the robust adaptive control(RAC) scheme. The plant parameters are assumed tobe unknown in this section. Section 4 gives the mathe-matical background of the recursive algorithm used toidentify the actuator dynamic. The experimental setupand simulations/real-time results are described and pre-sented in Sect. 5. Finally, concluding remarks are drawnin Sect. 6.

2 Modeling considering saturation

The basic design concept is presented in the frame-work of an n-link arm control problem. In vector form,the nonlinear differential equation describing the inte-grated actuator and manipulator dynamics are given asfollows [26]

D(q)q+C(q, q)q+G(q)+F(q)+Tl(t)=τl (1)

Jmd2θ

dt2 + Bmdθ

dt+ rτl = Kmi (2)

Ldi

dt+ Ri = V − Kb

dθ

dt, (3)

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Real-time robust adaptive control of robots

where q ∈ �n denotes the generalized coordinates rep-resenting joint position, D(q) ∈ �n×n is a symmetricpositive-definite inertia matrix, C(q, q) ∈ �n×n is thecoriolis and centrifugal force matrix, G(q) ∈ �n×1

represents the gravitational force vector, F(q) ∈ �n×1

is the static and dynamic friction terms, Tl(t) ∈ �n×1

is the vector of disturbance torques, Jm ∈ �n×n is adiagonal matrix of the lumped actuator rotor inertiasalong with the motor side elements of the transmis-sion, θ ∈ �n×1 is motor shaft angles, Bm ∈ �n×n isdiagonal matrix of the lumped actuator damping coef-ficients and motor side elements of the transmission,τl ∈ �n×1 is the load torques expressed on the loadside of the transmission, Km ∈ �n×n is diagonal matrixof motor-torque constants, i ∈ �n×1 is armature cur-rents, L ∈ �n×n is a diagonal matrix of the armatureinductances, R ∈ �n×n is diagonal matrix of armatureresistances, V ∈ �n×1 is armature voltages applied forthe joint actuators, and Kb ∈ �n×n is diagonal matrixof the back EMF constants, and r is an n × n trans-mission matrix, which maps the n × 1 vector of motorcoordinate velocities, dθ/dt , into the vector of gener-alized joint coordinate velocities, dq/dt , as

dq

dt= r

dθ

dt(4)

By the last transformation, the actuator dynamics in therobot side elements are then given by

Jmr−1 d2q

dt2 + Bmr−1 dq

dt= Kmi − rτl (5)

Ldi

dt+ Ri = V − Kbr−1 dq

dt. (6)

It is worth noting that in all practically well-designedactuators, the armature inductance L is small. Thus,for small enough L > 0, Eqs. (5) and (6) govern asingularly perturbed system, in which i and q are thefast and slow variables, respectively [27]. By settingL = 0 in (6), solving for i , and substituting into (5), thefamiliar two-order differential equation of the actuatoris obtained as follows:

Jd2q

dt2 + Bmdq

dt= V − RK −1

m rτl (7)

where

J = RK −1m Jmr−1 , Bm =

(RK −1

m Bm + Kb

)r−1.

Now, combining Eqs. to (1) and (7) yields the followingoverall dynamic of electrically driven robot:

d

dt

[qq

]=

[q−M−1(q) (N (q, q)q+H(q, q)− V )

],

(8)

where

M(q) = J + RK −1m r D(q)

N (q, q) = Bm + RK −1m rC(q, q)

H(q, q) = RK −1m r (G(q)+ F(q)+ Tl(t)) .

As can be seen from (8), it involves strong cou-pling between the joint motions. Thus, multivariablecontroller design for such highly nonlinear dynam-ics is generally complex and computationally diffi-cult. Moreover, from practical implementation pointof view, Eq. (8) suffers from control signal constraintthat requires nonlinear approaches in modeling, identi-fication, and control [28]. Toward this end, we assumedthat, the range of motor input voltage (V ) is constrainedto some certain voltage range called motor saturation.This saturation usually occurs between output of themotor controller and the PWM module. Let � denotesthe voltage space, defined as

� = V ∈ �n : |Vi | < V maxi , {i = 1, . . . , n} , (9)

where V maxi > 0 is the maximum allowable voltage for

the ith joint. By this definition, Eq. (8) can be writtenin the following form:

d

dt

[qq

]=

[q

V +�(q, q, q)

], (10)

where

�(q, q, q) = (I − M(q)) q − N (q, q)q − H(q, q),

The motivation behind our new formulation is that theeffect of parameter variations, external disturbances,and the common nonlinearities caused by the actuatorcan be lumped in a single term �(q, q, q). Thus, ourmain challenge is to maintain stability of (10) withoutusing any information of the nonlinearities and uncer-tainty bounds, except that they are bounded.

3 Adaptive control scheme

3.1 System modeling

Industrial robots are typically overdesigned in the senseof heavy and consequently rigid links with high-gear

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transmission mechanisms. As a result, they can bedescribed by linear and decoupled dynamic systems.Therefore, the control implementation for an n-DOFrobot manipulator can be done by n independent con-troller via independent joint control strategy in paral-lel. To do this, it can be easily shown that the nonlineardynamic model of a robot with n-DOF can be consid-ered as n decentralized subsystems, such that we cancontrol each of DOF separately. See Appendix 1 forfurther Information. In order to develop RAC, supposethat Eq. (10) for the ith joint can be described by thefollowing state and output equations:

x = Ax + BVi + � (11)

y = Cx, (12)

where the state vector x = [qi qi ]T denotes the mea-surable states of the system, y is the joint position,(A, B) is assumed to be controllable, and

A =[

0 10 0

], B =

[01

], � =

[0

�i (q, q, q)

],

C = [1 0

],

all have appropriate dimensions. Equation (11) is a dis-turbed double integrator with two poles in s-plane ori-gin. Thus, we should first stabilize the assumed model(11)–(12) by using a linear state feedback control asfollows:

Vi = −K x + K0ν, (13)

where ν is new control input for the ith joint, and Kand K0 are positive state feedback and zero frequencygain index, respectively. Now, substituting (13) into(11) yields

x = (A − BK )x + BK0ν + �. (14)

In advance, we assume that the following constructionof closed-loop system is given, which guarantees someperformance requirements and the stability criterions ofthe closed-loop system without considering saturation

xd = (A − BK )xd + BK0νd (15)

yd = Cxd (16)

with νd and yd denoting the reference trajectory anddesired output in the joint space. Now, suppose the errorvariables are defined as

e = x − xd, ξ = y − yd , ℵ = ν − νd (17)

that yields

e = (A − BK )e + BK0ℵ + � (18)

ξ = Ce. (19)

Now, the control problem can be stated as designing acorrective control input ℵ such that the input voltage isconstrained to the voltage space (9), and the joint dis-placements q asymptotically converge to the desiredjoint displacement qd. It must be emphasized that, thedevelopment of the proposed control Law is under theassumption that complete information of the actuatorand robot dynamic is not available (i.e., we have not anyknowledge of parameters plate or data sheet which issupplied usually by the manufacturer, except the volt-age range). To achieve this, the following variables aredefined

δ = e(p) −p∑

j=1

b j e(p− j) (20)

= ℵ(p) −p∑

j=1

b jℵ(p− j), (21)

where b j s are zero or positive real scalars to bedesigned. The core idea of these definitions is thatthe perturbation can be approximated by the follow-ing ordinary differential equation:

�(p) =p∑

j=1

b j�(p− j). (22)

This approximation opens the possibility to obtain thecorrective input ℵ to include it into the controller tocompensate the perturbation present in the plant.

Remark 3.1 Equation (22) is a p order ordinary differ-ential equation. It can be easily shown that the solutionof this equation is a continuous function as

�(t) =p∑

i=1

ci eλi t cos (ωi t + θi ) . (23)

It is interesting to investigate the capability of the lastassumption, Eq. (22), from a function approximationcapability point of view. Herein, we will prove thatEq. (23) has the universal approximation capability. Inthe following we suppose that the input universe ofdiscourse T is a convex set in �n . First we need thefollowing useful theorem.

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Stone–Weierstrass Theorem [29]Let � be a set of real continuous functions on a

convex set T . If

1. � is algebra, that is the set � is closed under multi-plication, addition, and scalar multiplication;

2. � separates points of T , i.e.

∀t1, t2 ∈ T , t1 �= t2 , ∃ �(t) ∈ � : �(t1) �= �(t2).(24)

3. � vanishes at no point of T , that is,

∀t ∈ T, ∃ �(t) ∈ � : �(t) �= 0 (25)

then for any real continuous function �(t) on T andarbitrary ε > 0, there exists a function f (t) in �

such that

Supt∈T

| f (t)− �(t)| < ε. (26)

Proposition 1 (Universal Approximation Theorem)Let �(t) be a continuous real function on a convex

set T in �n. Then for each arbitrary ε > 0, there existsa function in the form of

p∑i=1

ci eλi t cos (ωi t + θi ) (27)

such that

Supt∈T

∣∣∣∣∣p∑

i=1

ci eλi t cos (ωi t + θi )− �(t)

∣∣∣∣∣ < ε. (28)

The proof can be found in Appendix 2 It should be noted that proposition 1 proves only

existence result. That is, we can always find p andb j coefficients and therefore ci , ωi , and λi which canapproximate any continuous function with arbitrarysmall accuracy, such that, assumption (22) can be sat-isfied. In other words, it implies that there exist p andb j coefficients that validate Eq. (22) for disturbancedynamic approximation. However, appropriate valuesof order p and differential equation coefficients b j areto be determined mainly through trial and error relatedto application’s type. For our systems under study, it hasbeen shown that even a small p = 1 yields good results.

The validity of the proposed approach will be verifiedwith the experimental results and observations. Attendto linear system properties and taking time derivativesof both sides of (18) we arrive at

δ = (A − BK ) δ + BK0. (29)

Now, we define a coordinate transformation as

z = [ξ ξ . . . ξ (p−1) δT

]T ∈ �(p+2)×1 (30)

that gives the following system state-space equation innew coordinates

z = z +�

y = H z, (31)

where is new single control input and

=

⎡⎢⎢⎢⎢⎢⎢⎢⎣

0 1 0 · · · 0 00 0 1 · · · 0 0...

.

.

....

.

.

....

.

.

.

0 0 · · · · · · 1 0bp bp−1 · · · · · · b1 C0 0 · · · · · · 0 A − BK

⎤⎥⎥⎥⎥⎥⎥⎥⎦

∈ �(P+2)×(P+2)

� = [0 . . . 0 (BK0)

T]T ∈ �(P+2)×1,

H = [1 0 · · · 0

] ∈ �1×(P+2). (32)

3.2 MRAC mechanism

Making use of the definitions outlined in Sect. 3.1, thedesign procedure is now to search for an adaptive con-troller as

= −μz = −μ0δ −p∑

j=1

μ jξ(p− j) (33)

through an improved Lyapunov-based MRACapproach, so that all the signals in the closed-loop sys-tem are bounded and the plant output tracks the refer-ence model as close as possible [30]. Now, to proceedwith subsequent development and closed-loop stabilityanalysis, the following definitions are presented.

Definition 3.1 For given controller (33), we achievethe closed-loop system as

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z =

⎡⎢⎢⎢⎢⎢⎢⎢⎣

0 1 0 · · · 0 00 0 1 · · · 0 0...

......

......

...

0 0 · · · · · · 1 0bp bp−1 . . . . . . b1 C

−BK0μp −BK0μp−1 · · · · · · −BK0μ1 A − BK − BK0μ0

⎤⎥⎥⎥⎥⎥⎥⎥⎦

z, (34)

where μ ∈ �1×(p+2), μ0 = [μ0,1 μ0,2

] ∈ �1×2,and μ j ∈ �{ j = 1, 2 . . . , p} are tuning parameterswhich should be designed such that the closed-loopsystem be Hurwitz.

Definition 3.2 The “reference model,” which embod-ies the performance of the desired tracking error, isassumed to be as

zm = m zm (35)

where zm ∈ �(p+2)×1 is the desired tracking error vec-tor defined by (30), and

m =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

0 1 0 · · · 0 0

0 0 1 · · · 0 0...

......

......

...

0 0 · · · · · · 1 0

bp bp−1 · · · · · · b1 C

−Bαp −Bαp−1 · · · · · · −Bα1 A − Bα0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(36)

is Hurwitz with α j ∈ �, { j = 1, 2 . . . , p} and α0 ∈�1×2. The reference model (36) is stable; so there existsa unique symmetric positive-definite matrix �, Q suchthat

Tm� + � m + Q = 0, (37)

where �, Q ∈ �(P+2)×(P+2) are symmetric, con-stant, and positive-definite matrices with appropriatedimensions.

Definition 3.3 It is supposed that adaptation error vec-tor is defined as

E = zm − z. (38)

Hence, the error state-space equation is as follows:

E = m E + ψz, (39)

where

ψ =

⎡⎢⎢⎢⎢⎢⎢⎢⎣

0 0 · · · 00 0 · · · 0...

... · · · ...

0 0 · · · 00 0 · · · 0

B(K0μp − αp

) · · · B (K0μ1 − α1) B (K + K0μ0 − α0)

⎤⎥⎥⎥⎥⎥⎥⎥⎦

∈ �(P+2)×(P+2). (40)

Details of the discussion on accomplished calcula-tion are omitted here due to space limitation.

Theorem 3.1 Consider the closed-loop system stateequation given by (34). It is supposed that definitions3.1, 3.2, and 3.3 are satisfied, and the adjustable para-meter are tuned by

μ j (t) = K −10

[ρ j

d

dt

(ξ (p− j)(t)ϑ(t)

)

+ ξ (p− j)(t)ϑ(t)/ρ j

], { j = 1, 2 . . . , p}

μT0 (t) = K −1

0

(ρ0

d

dt(δ(t)ϑ(t))T + ρ−1δ(t)ϑ(t)

)

(41)

where ρ j and ρ j are arbitrary positive and non-negative scalars, ρ and ρ0 = diag

[ρ0,1, ρ0,2

]are the

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2×2 gain matrices with positive and non-negative diag-onal elements, and ϑ(t) is weighted error defended by

ϑ(t) = �p+2z (42)

in which �p+2 denotes the end row of matrix �. Underthese conditions, the solution tends to zero as time goesto infinity, while the entire signal remains bounded.

Proof Consider a Lyapunov function candidate

V = ET�E +⎡⎣

p∑j=1

ρ j(K0μ j − α j − μ j

)2

⎤⎦

+ [K + K0μ0 − α0 − μ0]

×ρ [K + K0μ0 − α0 − μ0]T , (43)

where μ and μ j are functions in respect of time to bespecified. From the Eq. (39) we have

V = −ET QE + 2p∑

j=1

ρ j[K0μ j − α j − μ j

]

× [K0μ j − ˙μ j

] + 2zTψT�E

+ 2 [K + K0μ0 − α0 − μ0] ρ[K0μ0 − ˙μ0

]T.

(44)

By applying the adaption law (41), substituting (30),(38), (40) into (44) and using some mathematical cal-culations we have

V = −ET QE − 2p∑

j=1

μ jξ(p− j)ϑ − 2μ0δϑ (45)

Now, because the time derivative of V should be anegative-definite function, we choose μ j and μ0 as fol-lows:

μ j = ρ jξ(p− j)ϑ, μ0 = (δϑ)T ρ0 (46)

By noting (46), (45) can be rewritten as

V = −ET QE − 2p∑

j=1

ρ j

(ξ (p− j)ϑ

)2

− 2 (δϑ)T ρ0δϑ ≤ 0 (47)

Therefore e(t) → 0 as t → ∞. Moreover the adaptivescheme is asymptotically stable as well. In other word,by using the proposed adaptive controller, the states ofthe system track the desired states asymptotically, andthe closed-loop stability will be achieved.

Remark 3.2 For robotic application, the initial condi-tions of the desired and actual joint trajectory are oftenthe same, that is, the initial state for the desired trajec-tory can be set to be as same as the actual one. Thisyields a zero initial error

zm(0) = 0 (48)

with a small initial control signal. Thus, the solution of(35) is zero as

zm(t) = zm(0)e m t = 0 t ≥ 0. (49)

Under this assumption and considering asymptoticallystability of the adaptation error, z converges to zm .When the corrective control input ℵ generated by (21),(33) and (41), it needs to be translated into the actualcontroller output ν as

ν = νd + ℵ. (50)

The main contribution of this mapping is as follows.Once, the controller output exceeds the actuator lim-its, the error between the desired and measured outputincreases. This error signal is utilized for adjusting theadaptive controller parameters (41), (42), and the cor-rective control input. Now, by tracking the improvedtrajectory ν, the control system does not experiencesaturation for a long time in spite of existing satura-tion nonlinear operator and other uncertainties effect.In other words, what is basically happens here is keep-ing the behavior of the system as close as possible tothe behavior of the desirable system given by (15) and(16). It should be noted that, this purpose is achievableby selecting a proper model reference and coefficientsb j such that the bandwidth of the outer robustness loopcovers the frequency range of operation of the innerfeedback controlled system. The control structure isshown in Fig. 1. Now, by a simple block manipulationand some mathematical calculations it can be easilyshown that the scheme represented in Figs. 1 and 2 isequivalent.

In order to show this equivalency, Laplace transformof (14) and (15) under zero initial conditions are givenby�

X = (SI − A + BK )−1(

BK0�ν +��

)(51)

BK0�ν d = (SI − A + BK )

�

X d, (52)

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Fig. 1 Block diagram ofthe robust adaptive controlscheme

Fig. 2 The equivalentschematic of robust adaptivecontrol scheme

where (�•) shows Laplace transform of the time domain

variables. Furthermore, by using (50) we have

BK0�ν = BK0

�ν d + BK0

�ℵ, (53)

where�ℵ is designed for minimizing the tracking error.

Combining (21) and (33) and taking Laplace transformof them under zero initial condition yield

�ℵ = −�μ ∗�z(

S p −p∑

j=1b j S p− j

) , (54)

where * denotes convolution operator. It should be

emphasized that both�μ and

�z are functions in respect

of the output error and its derivative. Now, multiplica-tion both sides of (54) by BK0 gives

BK0�ℵ = −BK0

�μ ∗�z(

S p −p∑

j=1b j S p− j

) (55)

With these state of meanings and using (51)–(55), theRAC scheme is complete. The equivalent block dia-gram of the proposed scheme is presented by the blockdiagram of Fig. 2. As can be seen, the proposed RAC isequivalent to a Feedforward control structure, consid-ering to uncertainties/time-varying parameters, exter-nal disturbance, and control input constraint. Here,(52) constitutes the Feedforward path, based on a lin-

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ear feedback controlled simple model. It is free fromactuated manipulator dynamic and does not need toinversion model of saturation nonlinearity for Feedfor-ward path implementation. The validity of the proposedapproach will be discussed in accordance with the sim-ulations/experimental results and observations.

Remark 3.3 Contrary to other robust and adaptive con-trol strategies that require a large number of basis func-tion and consequently large weighting matrices due tohave a good approximation of system, [31–34], rea-sonable tracking accuracy can often be achieved withrelatively low uncertainty model error (p = 1 or 2)[35,36].

4 Identification of the plant model

In this section, a closed-loop parametric identificationtechnique, based on classic parameter estimation meth-ods, was applied to identify the actuator dynamic. Theintegrated model of the identified actuator and mechan-ical subsystem is then applied for simulation processin the next section. The identification procedure is asfollows [37]:

4.1 Selection of the model structure

Let the input voltage and the angular position of theaxis be related as

A(q)y(t) = B(q)u(t), (56)

where q is the forward shift operator, and A(q) andB(q) are the polynomials

A(q) = q2 + a1q + a2, B(q) = b0q + b1.

4.2 Experimental planning

For the determination of the transfer function parame-ters, a classic closed-loop identification technique isapplied to the real plant. This is useful because the plant(56) is non-self-regulated in open loop conditions. Notethat there are no reasons to use a specific controller forthe identification procedure. However, for this exper-imental system, a self-tuning regulator (STR) wasused.

The experiments were carried out in tracking, con-sist of a sequence of step changes as the set point withenough time separation to reach the steady state. It oper-ates such that a voltage signal coming from controlleroutput is used to excite the system, and the motor’sangular position is measured. This data pattern is trans-ferred to PC via A/D converters and applied in the esti-mation algorithm in order to identify the actuator para-meters and ultimately solving the Diophantine equationdue to controller tuning. Figure 3 shows a schematicdiagram for the identification process.

4.3 Parameter estimation

An online algorithm for estimation of the parametersof the actuator, based on a recursive least square (RLS)method, was used in this work. Suppose that, Eq. (56)can be written as the following difference equation:

y(t) + a1 y(t − 1) + a2 y(t − 2) = b0u(t − 1)

+ b1u(t − 2). (57)

Fig. 3 Schematic diagramfor the identification process

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Introduce the parameter vector

� = [ b0 b1 a1 a2 ]T (58)

and the regression vector

φT (t − 1)=[ u(t − 1) u(t − 2) −y(t − 1) −y(t − 2) ]which its elements are the delayed input and output.The model can formally be written as the regressionmodel

y(t) = φT (t − 1)�. (59)

Now, the least square estimator is given by [38]

�

�(t) = �

�(t − 1)+ α(t)ε(t)

ε(t) = y(t)− φT (t − 1)�

�(t − 1)α(t)= P(t − 1)φ(t − 1)(1+φT (t−1)P(t−1)φ(t−1))−1

P(t) = (I − α(t)φT (t − 1))P(t − 1),

(60)

where P(t) is a positive-definite matrix initialized tobe cI (I : identity matrix, 100 < c < 10000) [39]. Thisyields the following transfer function:

B(q)

A(q)= 0.001306q + 0.001467

q2 − 1.299q + 0.299(61)

Transforming discontinuous model to continuous formunder Ts = 50 ms gives

T (s) = −0.01216s + 1.91

s2 + 24.15s + 1.201e − 12(62)

As can be seen, the RLS method used for identifica-tion yields an LTI but non-minimum phase model ofthe original nonlinear system. Considering the fact thatthe identification is done online in the closed-loop sys-tem through large step input changes, the model is anconservative low frequency approximate model of thenonlinear time variant system for a trajectory taken.Figure 4 shows the estimate of the parameters.

4.4 Nominal model validation

The actual output of the set-point control is comparedwith the simulated one in Fig. 5. The results presentedin this Figure acceptably validate the nominal model.

0 5 10 15 20 25-2.5

-2

-1.5

-1

-0.5

0

0.5

1

Time (sec)

Iden

tifie

d P

aram

eter

s

b0b1a1a2

Fig. 4 Parameter estimates for the coefficients of A(q), B(q)

0 5 10 15 20 25-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Time (sec)

Tra

ject

ory

Tra

ckin

g (r

ad)

ExperimentalReferenceSimulated

0 0.01 0.02-1

0

1

2x 10

-3

Fig. 5 Experimental and simulated responses of the controlledvariable

The initial undershoot for simulated response showsexistence of zero on the RHS of imaginary axis. Now,the integrated model of mechanical subsystem andidentified actuator is applied for simulation process, inthe next section. In this stage, by modeling the addedpayload as a point mass at the end of the arm, the val-ues of gravitational loading and inertia are calculatedas 0.25 N m and 0.0022 kg m2 respectively.

5 Real-time implementation

5.1 Experimental setup

The experimental setup using a two-link Elbow robotmanipulator is shown in Fig. 6. The first joint is drivenby geared permanent magnet DC motor characterized

123


Fig. 6 A schematic of theexperimental setup

by Barber-Colman Company operating within ±12 Vinput, and the second link is released. The first motorshaft is equipped by a potentiometer and a Tacho-generator. The measured input–output data are trans-ferred to the computer (Pentium II 366 MHz) by adata acquisition card (ADVANTECH PCL-818L withup to 100 kHz sampling rate, 12 bit high speed A/Dconverter with a conversion rate of max 40 kHz). Thedata acquisition card permits us to control the prac-tical robot through user-defined programs in MAT-LAB/SIMULINK environment. The sampling intervalof the data acquisition process is set to 5 ms. The sam-pling period is less than 10 % of the step responsesettling time for adequate recognition of the systemdynamics [40]. Furthermore, a low-pass first-order fil-ter with a cutoff frequency of 10 Hz is used for poten-tiometer noise mitigation. From the view point of con-troller implementation, the natural question is the possi-ble implementation of our results for robots with higherDOF. Since, the proposed control system structure islinear and decoupled, and the interactions are consid-ered as part of uncertainties, then the control algo-rithm for an n-DOF robot manipulator could be imple-mented by n independent controller via independentjoint control strategy in parallel. Therefore, the selectedcase study results can be extended to wide rangeof practical robots with higher DOF without loss ofgenerality.

5.2 Experimental results

For the first test, a unit step input is used to generate thereference trajectory in the joint space. The joint mustmove from 0 radian to +1 rad in short time with lowovershoot. The following values for the state feedbackvector K , the order of disturbance model p, and theadaptive control parameterμwere found to award goodresults

K = [64 16

], p = 1, b1 = 0

μ1(t) = 0.03ξ(t).ϑ(t)+ 0.1∫ t

0ξ(τ ).ϑ(τ ).dτ + 1

μT0 (t)=

[0.05 00 10−4

] (δ(t)ϑ(t)+

∫ t

0δ(τ ).ϑ(τ ).dτ

)+b,

(63)

where

ϑ(t) = 10−3e(t)+ e(t)+17e(t), b = [1.34 10−3

]T.

(64)

In order to illustrate the performance of the proposedcontroller under large external disturbances that canpotentially cause saturation, the following disturbanceimposed to the control signal V .

w(t)=20u(t − 4)−10u(t − 10)+5u(t − 16) (65)

where u(t) indicates the unit step function. Such astrong disturbance is quite likely to move the actua-tor to the saturation zone. It should be emphasized that(65) can be representative of additive uncertainties aswell as usual output disturbances before the saturationblock. By this assumption, Fig. 7 shows the trajectorytracking of the joint in the presence of external distur-bances. As can be seen from Fig. 8, the tracking errorfor the proposed controller occurs only at the jump timeand vanishes rapidly in spite of saturation of the con-trol signal by the saturation element, while this errorincreases in the absence of exterior-loop adaptive con-troller. The technical limits such as motor voltage, cor-rective control signal, and adapted control gains arealso shown in Figs. 9, 10, and 11, respectively. Fig-ure 10 indicates that ℵ constitutes a majority portionof ν. It highlights importance of the proposed con-trol law in reduction of tracking errors by tracking theimproved trajectory injected to the inner loop. There-fore, the proposed control scheme provides reasonable

123


0 2 4 6 8 10 12 14 16 18 200

0.2

0.4

0.6

0.8

1

1.2

1.4

Time (sec)

Tra

ject

ory

trac

king

(ra

d)

Without RAC

RACRAC (identified model)

Fig. 7 Desired and actual system response

0 2 4 6 8 10 12 14 16 18 20-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

Time (sec)

Tra

ckin

g er

ror

(rad

)

Without RAC


Fig. 8 Tracking error subject to external disturbance

performance and robustness for the two-link manipula-tor in the presence of external disturbance, un-modeleddynamics, and external load torque. To provide repro-ducibility feature, the simulation results have also beenprovided based on identified model.

As a further test, we present experimental results fortracking of a cycloidal (bow-shaped) path, given by

xd(t) ={

0.25 (0.2π t − sin (0.2π t)) 0 ≤ t ≤ 10π2 10 < t

(66)

The control parameters are selected as

K = [200 20

], p = 1, b1 = 0

0 2 4 6 8 10 12 14 16 18 20-15

-10

-5

0

5

10

15

Time (sec)

App

lied

volta

ge (

volt)

Without RAC


Fig. 9 Applied voltage subject to external disturbances

0 2 4 6 8 10 12 14 16 18 20-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Time (sec)

Impr

oved

/Aux

illar

y co

ntro

l sig

nals

(ra

d) Improved trajectory

Auxillary inputImproved trajectory/identified model

Auxillary input/identified model

Fig. 10 Auxiliary and improved trajectory

0 2 4 6 8 10 12 14 16 18 20-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Time (sec)

Ada

ptiv

e co

ntro

l par

amet

ers

µ1

µ0,1

µ0,2

µ1/identified model

µ0,1/identified model


Fig. 11 Variation of the adjustable parameters

123


0 2 4 6 8 10 12 14 16 18 200

0.5

1

1.5

2

2.5

Time (sec)

Tra

ject

ory

trac

king

(ra

d)

Without RAC

Desired trajectoryRAC

RAC (identified model)

Fig. 12 Desired and actual system responses

0 2 4 6 8 10 12 14 16 18 200

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

Time (sec)

Tra

ckin

g er

ror

(rad

)

Without RAC


Fig. 13 Tracking error subject to external disturbance

μ1(t) = 20ξ(t)× ϑ(t)+ 20∫ t

0ξ(τ )× ϑ(τ)× dτ

μT0 (t) =

[10 00 10−3

] ∫ t

0δ(τ )× ϑ(τ)× dτ

+[

1 00 10−3

]δ(t)ϑ(t) (67)

where

ϑ(t) = 10−3e(t)+ e(t)+ 10e(t) (68)

As shown in Fig. 12, the joint should move from 0radian to +2 rad in 10 s. One second period of 10 s dura-tion follows the reference values being held constant toshow the regulation performances of the proposed con-troller. To test the robustness of the proposed approach,

0 2 4 6 8 10 12 14 16 18 20-6

-4

-2

0

2

4

6

8

10

12

Time (sec)

App

lied

volta

ge (

volt)

Without RAC


Fig. 14 Applied voltage subject to external disturbances

0 2 4 6 8 10 12 14 16 18 20-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Time (sec)

Impr

oved

/Aux

illar

y co

ntro

l sig

nal (

rad)

Improved trajectory

Auxillary inputImproved trajectory/identified model

Auxillary input/identified model

Fig. 15 Auxiliary and improved trajectory

a unit step disturbance is applied to the robot manipu-lator at t = 4 s. Figures 12 and 13 show the tracking tra-jectory and tracking error for the joint in the presenceof external disturbances. It can be seen from Fig. 13that the maximum tracking error is less than 0.1 radwhich is damped fast. The actuator voltage, the con-trol input ν /corrective control ℵ, and adaptive controlgains are also plotted in Figs. 14, 15, and 16. As appearsin Fig. 14, there is a jump in the control signal due toremoving the injected external disturbance, which isquite logical. Simulation results for both experimentaland identified model show that there is virtually a lit-tle change in the responses of the RAC system. Thus,the proposed control scheme is successful in rejectingdisturbances and some excluded nonlinearities in the

123


0 2 4 6 8 10 12 14 16 18 20-0.5

0

0.5

1

1.5

2

2.5

Time (sec)

Ada

ptiv

e co

ntro

l par

amet

ers

µ1

µ0,1

µ0,2

µ1/identified model



Fig. 16 Variation of the adjustable control parameters

nominal plant model based on using a model-free con-troller design approach.

6 Conclusion

Many adaptive control techniques promise improve-ments of the performance of the control system underactuator saturation limitations. However, a majority ofthem does not guarantee asymptotic closed-loop sta-bility in the presence of significant external distur-bance which forces the system to exhibit more non-linear behaviors. Generally, any external disturbancecauses tracking error and actuator saturation prob-lem. Using a RAC approach for tracking trajectory ofrobot with actuator saturation effect was developed inthis paper. The proposed approach guarantees stabil-ity, ensures tracking and disturbance rejection capabil-ities while avoiding windup. Some experimental resultswere given to prove the capability of the proposedapproach.

Acknowledgments The authors would like to thank Dr Imanzamani for his comments on the article. They also thank Hosseinsalehian for his comments on the robotic mechanical design.

Appendix 1

In order to develop the proposed control scheme, it isconvenient to view each joint as a subsystem of theentire actuated manipulator system. Toward this end,let V presented by

V = [V1 V2 · · · Vj · · · Vn

]T. (69)

By this definition, the dynamic model (10) can beexpressed by a collection of n second-order nonlineardynamic which is described by following scalar differ-ential equations:

mii (q)qi + ni (q, q)qi + hi (q, q)+n∑

j=1, j �=i

mi j (q)q j

= Vi {i = 1, 2, . . . , n} , (70)

where the subscript “i” indicates the ith element, andmii (q) is the varying effective inertia seen at the ithjoint. Thus, Eq. (70) can be expressed as follows:

mii (q)qi + di (q, q, q) = Vi {i = 1, 2, . . . , n} ,(71)

where

di (q, q, q) = ni (q, q)qi +hi (q, q)+n∑

j=1, j �=i

mi j (q)q j

(72)

is treated as disturbance. di (q, q, q) is imposed to theith joint and contains the gravity, friction, coriolis, cen-trifugal torques for the ith joint, inertia coupling effectsfrom the other joints as well. From the system point ofview, (72) summarizes the coupling between the ithsubsystem and the remaining subsystem. Plus and sub-tracting qi from (71) and multiplying both sides of itby m−1

i i yields available model, presented in Sect. 3.

Appendix 2

Proof of propositioin 1 Let � to be a set of continuousfunction on T , and T is Convex set in the form of (23).Now, suppose �1(t) and �2(t) are given as

�1(t) =p∑

i=1

ci eλi t cos (ωi t + θi )

�2(t) =p∑

j=1

c j eλ j t cos

(ω j t + θ j

)(73)

123


we have

�1(t)+ �2(t) =p∑

i=1

ci eλi t cos (ωi t + θi )

+p∑

j=1

c j eλ j t cos

(ω j t + θ j

)

�1(t).�2(t) = 1

2

p∑i=1

p∑j=1

ci c j e(λi +λ j )t cos

((ωi + ω j

)t

+ (θi + θ j

)) + 1

2

p∑i=1

p∑j=1

ci c j e(λi +λ j)t

cos((ωi − ω j

)t + (

θi − θ j))

(74)

Hence, �1(t)+�2(t) and �1(t).�2(t) ∈ h. Finally, forany arbitrary σ ∈ � we can get

σ.�(t) =p∑

i=1

σci eλi t cos (ωi t + θi ) (75)

which is also in the form of (23). So, by considering(74) and (75) we can conclude that � is algebra. There-fore, the first condition of Stone–Weierstrass Theoremis satisfied for �. We show that � separates points onT . We choose the parameters of the �(t) in the form of(23) as follows:

c1 = 1, λ1 = −1, ω1 = 0, θ1 = 0. (76)

Since t1 �= t2, then e−t1 �= e−t2 and therefore the sec-ond condition is also verified. To show that � vanishesat no point of T , we simply observe that any system inthe form of (23) with ωi = 0, θi = 0, and ci > 0 hasthe property of

∀t ∈ T, �(t) > 0. (77)

Hence, � vanishes at no point of T . Thus the threeconditions of the Stone–Weierstrass theorem are satis-fied. Therefore the result follows by Stone–WeierstrassTheorem.

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