2616 IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 24, NO. 6, DECEMBER 2019

Mobile Robot ADRC With an AutomaticParameter Tuning Mechanism via Modified

Pigeon-Inspired OptimizationXingshuo Hai , Zili Wang, Qiang Feng , Member, IEEE, Yi Ren , Binghui Xu, Jingjing Cui ,

and Haibin Duan , Senior Member, IEEE

Abstract—This article proposes an enhanced active dis-turbance rejection control (ADRC) method for the attitudedeformation system of a self-developed mobile robot inconjunction with evolutionary game theory-based pigeon-inspired optimization (EGPIO). To achieve precise controlof the object, EGPIO is employed as an automatic tuningmechanism for ADRC, and simultaneously improves therapidity and robustness of the controller. In this novel algo-rithm, individuals in pigeon-inspired optimization (PIO) ad-just their strategies dynamically as the evolutionary gameproceeds so as to promote the adaptability of the originalPIO. Thus, the optimal solutions calculated by EGPIO willbe adopted as the main parameters of ADRC to control themobile robot. The effectiveness of the novel controller isvalidated by a series of simulations, including cases withnormal conditions and situations with disturbances. Exper-imental results indicate that the scheme we designed pos-sesses better control performances and shows attractiveadvantages for efficiency and fault tolerance compared withconventional ADRC.

Index Terms—Active disturbance rejection control(ADRC), evolutionary game theory (EGT), fault tolerance,mobile robot, pigeon-inspired optimization (PIO),robustness.

I. INTRODUCTION

THE large diffusion of control techniques witnessed inrecent years resulted in the spread of high stability and

reliability of mobile robots that belong to a typical mechatronicsystem [1]–[3]. With the increasing complexity of both theinside structures and outside environments of robotic systems,

Manuscript received January 4, 2019; revised August 19, 2019 andNovember 1, 2019; accepted November 11, 2019. Date of publicationNovember 13, 2019; date of current version December 31, 2019. Rec-ommended by Technical Editor H. R. Karimi. This work supported by theNational Pre-Research Foundation of China under grant 61400020109.This work was supported in part by the Laboratory of Hexability In-tegrated Product Design and Simulation, Beihang University, China.(Corresponding author: Qiang Feng.)

X. S. Hai, Z. L. Wang, Q. Feng, Y. Ren, B. H. Xu, and J. J. Cuiare with the School of Reliability and Systems Engineering, BeihangUniversity, Beijing 100191, China (e-mail: haixingshuo@buaa.edu.cn;wzl@buaa.edu.cn; fengqiang@buaa.edu.cn; renyi@buaa.edu.cn;xubinghui@buaa.edu.cn; cuijingjing@buaa.edu.cn).

H. B. Duan is with the School of Automation Science and Elec-trical Engineering, Beihang University, Beijing 100191, China (e-mail:hbduan@buaa.edu.cn).

Color versions of one or more of the figures in this article are availableonline at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TMECH.2019.2953239

disturbances are known to occur variably at unknown times.Thus, a control system of mobile robots is required to possessfault tolerance so that it can support reliable operation in thepresence of uncertainties and disturbances [4], [5].

Currently, numerous fault tolerance strategies for enhanc-ing the robustness of controllers for mobile robots have beensubstantially developed, such as the robust control [6], [7],H-infinity control [8], [9], adaptive control [10], [11], sliding-mode control [12], [13], and active disturbance rejection control(ADRC) [14], [15]. Taking robustness as the objective of aclosed-loop system, robust control techniques [16] and theirimproved versions, H-infinity controllers have been shown toprovide higher rejection capabilities for disturbances and faults.However, the control structure is complicated, and tuning thelarge number of controller parameters is difficult. Though anH-infinity controller considers more uncertainties of the systemand relaxes the parameter perturbation space, implementationis also difficult due to the need for a reasonably precise model.Compared with robust control methods, the robustness in adap-tive control is always included as a part of the solution andalleviates the need for a priori knowledge about the boundson uncertain parameters [17]. However, the ideal results areobtained at the cost of a large amount of computation. As non-linear control paradigms, both sliding-mode control and ADRCwere developed to deal with uncertainties. The former approachdynamically adjusts a model according to the current state ofthe system and makes it insensitive to disturbances [18], whilethe latter approach treats unknown dynamics and external dis-turbances as an extended state of a plant that is compensated bya real-time control law [19]. Nevertheless, the lack of finite-timeconvergence and the existence of large chattering have becomethe main obstacles for sliding-mode control.

Based on its control design, ADRC is easily implementedand requires very little information about plant dynamics [20]. Inmany engineering fields with complex systems, ADRC performsremarkably well since the complexity can be weakened bygrouping all the nonlinearities and external disturbances into ageneralized term [21], [22]. Further, with the theoretical progressin the study of the stability and convergence of the ADRCmethod [23]–[25], it has found wide application in mechatronicsystems [26], [27].

Most works utilizing an ADRC controller put more empha-sis on model construction, stability analysis and performance

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HAI et al.: MOBILE ROBOT ADRC WITH AN AUTOMATIC PARAMETER TUNING MECHANISM 2617

evaluation. The parameter selection and adjustment of thecontroller are not elaborated since a trial-and-error tuningmethod is generally adopted in the control scheme. To deliveran appropriate control signal, a great deal of time will be spenton manual tuning of parameters because they play a significantrole in the system response performance. However, the processof manually tuning ADRC parameters is inefficient and time-consuming, especially for robotic systems in intelligent applica-tion backgrounds. Furthermore, due to differences in knowledgeand experience, achieving the optimal performance is difficultto guarantee. Therefore, it is indispensable to determine aneffective tuning mechanism of ADRC parameters to improveefficiency.

Pigeon-inspired optimization (PIO) algorithm [28] has provento be feasible and effective for optimization problems and man-ifests positive results [29]–[32]. Nevertheless, because of thefocus on simulation results, a critical issue has been neglectedduring the procedure, i.e., the separate operators in a PIO algo-rithm and most of its variants require man-made manipulationsof the number of iterations for different problems, limitingboth the adaptability and rapidity of the algorithm. To someextent, this operation is redundant, in particular, for real-timecontrol problems. The combination in [33] eliminates the needto set the number of iterations one after another, but linearlyupdated coefficients result in coordination between operatorsand coefficients being invariable. Thus, the algorithm may beinapplicable to nonlinear and complicated problems because theadaptability of the algorithm is limited by mechanical iteration.

Motivated by improving a PIO algorithm and applying thisalgorithm to automatically tune parameters in an ADRC con-troller, we introduce evolutionary game theory (EGT) [34] intobasic PIO, named EGPIO [35], and propose an efficient EGPIO-ADRC controller to enhance the fault tolerance capability ofa mobile robot with improvements in the proposed robustnesscontroller.

The contributions of this article are as follows. First, anEGPIO algorithm is proposed. Owing to this combination, theadaptability and rapidity of the original PIO have been greatlyimproved because the performance of the proposed approachdepends entirely on the solutions of a dynamic game instead ofempirically setting the main parameters for various problems.Second, an enhanced EGPIO-ADRC scheme is designed. Thenovel controller improves the efficiency of basic ADRC sinceparameters in the controller are determined based on the auto-matic tuning mechanism produced by EGPIO. Third, a hybridcontroller is developed for the attitude control of a self-designedmobile robot; the performance of this robot in terms of highrobustness, fast transient response, global optimization anddynamic adaptation is evaluated, and the effectiveness of theproposed control method is demonstrated through a series ofsimulations.

The rest of this article is organized as follows. Section IIgives the modeling scheme of a self-designed mobile robot.Section III outlines the control structure of the mobile robotbased on an ADRC controller. Section IV presents the detailsof the proposed EGPIO algorithm. Section V shows a prototypeof the proposed EGPIO-ADRC controller. Section VI explores

Fig. 1. 3-D simulation of the deformable intelligent robot.

Fig. 2. Control process of the deformable push rod in the motionsystem.

implementations and simulations of the developed concepts.Section VII concludes this article.

II. MODELING SCHEME OF A SELF-DESIGNED INTELLIGENT

MOBILE ROBOT

Considering the complexity of the environment and the spe-cific task requirements [36], a self-developed deformable mobilerobot, named “Dirgo,” was designed to achieve good mobil-ity and obstacle-crossing ability. In this section, the modelingscheme of the motion system in Dirgo is detailed.

A. Introduction of Dirgo

According to the basic criterion [37], the constitution of Dirgois composed mainly of a drive and energy system, a motionsystem, and a control system. To clearly show the composition,a three-dimensional simulation of Dirgo is shown in Fig. 1.

The motion system is the basis of Dirgo, including push roddeformation mechanisms installed at the four corners of thebody, four moveable wheels, a symmetrical pedrail, motors, etc.In the control system, an ADRC controller with an EGPIO algo-rithm is embedded into the device. As the main task of our work,the attitude control of Dirgo is accomplished by transformingthe positions of the moveable wheels. Therefore, when Dirgoencounters obstacles or disturbances, it can constantly adjust theattitude to reduce its inclination angle to improve the stability[38].

In this article, we concentrate on the motion system of Dirgoin which the core component is the push rod deformation instal-lation. A diagram of the control process of the deformable pushrod is shown in Fig. 2.

Initially, attitude angle signals are transmitted to the controllerbased on a special sensor. Thereafter, the information is analyzedand calculated by the controller and is transmitted to the actuator

2618 IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 24, NO. 6, DECEMBER 2019

Fig. 3. Schematic diagram of the mechanical structure of a deformablepush rod.

TABLE IPHYSICAL PARAMETERS OF THE MOTION SYSTEM

to drive the motor that controls the push rod. Based on the aboveprocedure, the rod connecting the moveable wheel and the fixedpoint will rotate around the fixed point to change the position ofthe moveable wheel. In this way, the coordination between thetwo moveable wheels further alters the attitude of Dirgo.

B. Mathematical Model of the Motion System

To establish a model in mathematical terms, a schematicdiagram of the mechanical structure of a deformable push rodin the motion system is first shown in Fig. 3. As this figureshows, the system is defined by two-degrees-of-freedom, i.e., themotion of the push rod expands along the axis and the motion ofthe joint rod rotates around the fixed point. The motor generatesinductive reactance R and drives the push rod with dampingb via an output gear with radius r1 and a telescopic device withelastic coefficient k1. Then, the joint rod connecting the fixedpoint and the moveable wheel rotates around the fixed pointunder the motion of the push rod.

The physical parameters of the motion system are presentedin Table I and were obtained from direct measurements of theprototype and a CAD model.

According to the mechanical model, the motion system canbe considered a single-input single-output system. The input isthe equivalent force FV of the motor input voltage, while theoutput is the rotation angle θ of the joint rod. Hence, the modelof the system can be expressed as

θ(s) = G(s) · FV (s) (1)

where G(s) is the transfer function from FV to θ. A closed-loopdiagram of the motion system can be further built, as shown inFig. 4.

Fig. 4. Closed-loop block diagram of the motion system.

Through analysis and calculation, the transfer function G(s)can be presented as

G(s) =

r(bs2 + k1s)

Rg2s4+b(g+R)s3+(gk1+Rk2r22+Rk1)s2+r2(k2bs+ k)

(2)

where r = r1r2, k = k1k2, g2 = r22M , and g = r2M . Subse-

quently, according to the method of inverse Laplace transform,the system dynamics in the time domain can be obtained as

g2Rθ(4) + b(g +R)θ(3) + (gk1 +Rk2r22 +Rk1)θ

+ r2(k2bθ + kθ) = rbFV + rk1FV (3)

where all coefficients are listed in Table I. Consider that thecoefficients of the fourth-order term in (3) by calculation aresmall enough to be neglected. According to the dynamic systemshown in (3), we have a simplified single-input single-outputdynamic system

θ(3) = f(θ, θ, θ, ω, t) + b0u (4)

where u and θ are the control signal and the system output,respectively, f(θ, θ, θ, ω, t) represents the total disturbances, ωdenotes the external disturbance, and b0 is the control signal co-efficient. By selecting the state variables as x1 = θ, x2 = θ, andx3 = θ, the proposed dynamic system in (4) can be expressed ina state equation as

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

x1 = x2

x2 = x3

x3 = f(x1, x2, x3, ω(t), t) + b0u

θ = x1.

(5)

III. ACTIVE DISTURBANCE REJECTION CONTROL OF THE

MOTION SYSTEM IN DIRGO

The ADRC controller consists of a tracking differentiator(TD), extended state observer (ESO), and nonlinear state errorfeedback (NLSEF) control law [21]. Based on (5), an equivalentthird-order ESO accompanied by the other components of theADRC controller acts on the motion system, and is shown inFig. 5.

Detailed descriptions are presented in the following sections.

A. Tracking Differentiator

The TD is used to track the reference signal v and configurethe transition process by its tracking signal v1 and its differential

HAI et al.: MOBILE ROBOT ADRC WITH AN AUTOMATIC PARAMETER TUNING MECHANISM 2619

Fig. 5. Control scheme of the motion system.

v2. The structure of the TD is given as follows:⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

e = v1 − v

fh = fhan(e, v2, r0, h)

v1 = v1 + hv2

v2 = v2 + h · fh

(6)

where e is the observer error and fhan is the optimal controlsynthesis function derived from discrete optimization theory.The iteration process of fhan(e, v2, r0, h) can be described by

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

d = r0h

d0 = r0h2

y = e+ hv2

a0 =√

d2 + 8r0 |y|

a =

{v2 +

(a0−d)2 sign(y), |y| > d0

v2 +yh , |y| ≤ d0

fhan = −{r · sign(a), |a| > d

r ad , |a| ≤ d

(7)

where r0 and r are the convergence rate coefficients, h is thestep length, and sign() denotes the sign function.

B. Extended State Observer

The ESO is designed to realize the accurate estimation ofsystem state variables x1, x2, and x3, and total disturbancesf(θ, θ, θ, ω) based on the control signal u and system output θ.In our scheme, an equivalent third-order ESO consists of threesecond-order ESOs connected in series as presented. The flowof the proposed ESO computation is given as follows:

⎧⎪⎨

⎪⎩

e1 = z1 − θ

z1 = z1 + h(z2 − β01e1)

z2 = z2 + h(−β02fe1)

,

⎧⎪⎨

⎪⎩

e3 = z3 − z2

z3 = z3 + h(z2 − β01e3)

z4 = z4 + h(−β02fe3 + b0u)⎧⎪⎨

⎪⎩

e5 = z5 − z4

z5 = z5 + h(z2 − β01e5 + b0u)

z6 = z6 + h(−β02fe5)

(8)

where fei = fal(ei, α, δ) = {ei

δα−1 , |ei| ≤ δ|ei|αsign(ei), |ei| ≥ δ

is a non-

linear function; z1, z3, and z5 are the estimated values of thestate variables x1, x2, and x3, respectively; z6 is the estimated

value of the total disturbances; α affects the adaptability to themodeling uncertainties and the disturbances; δ is the linear widthof the nonlinear function; β01 affects the estimation of the statevariables; β02 affects the estimation of the total disturbances;and b0 affects the value of compensation.

C. Nonlinear State Error Feedback Control Law

In NLSEF, a nonlinear function is utilized to approach the op-timal input of the system. The mathematical model is establishedas follows:

⎧⎪⎨

⎪⎩

e1 = v1 − z1

e2 = v2 − z2

u0 = −fhan(e1, ce2, r, h1)

(9)

where e1 and e2 are the state errors of the system, u0 is the errorfeedback control quantity, fhan() is given by (7), c andh1 are thedamping factor and accuracy factor, respectively, and r denotesa convergence coefficient.

Finally, the generation of the control signal u is realized bycompensating for the estimated values of the total disturbances,and is written as

u = u0 − z6

b0(10)

where the second term z6b0

represents the compensation of thetotal disturbances.

IV. EVOLUTIONARY GAME-BASED PIGEON-INSPIRED

OPTIMIZATION

A. Basic Pigeon-Inspired Optimization

There are two operators in PIO [28]. Given a popu-lation size N , the ith pigeon is defined by its positionXi = (xi1, xi2, . . . , xiD) and velocity Vi = (vi1, vi1, . . . , viD),and they will be updated in a D-dimensional search space ineach iteration t. The equation in the map and compass operatorcan be written as

Vi(t) = Vi(t− 1) · e−Rt + rand · (Xgbest −Xi(t− 1)) (11)

Xi(t) = Xi(t− 1) + Vi(t) (12)

where R is the map and compass factor, rand represents arandom number within [0, 1], and Xgbest represents the currentglobal best position.

In the landmark operator, the number of pigeons decreases byhalf to abandon unsatisfied pigeons. For a fitness function F (),the updating rules are defined as

Np(t) =Np(t− 1)

2

C(t) =

∑Xi(t) · F (Xi(t))

Np ·∑

F (Xi(t))(13)

Xi(t) = Xi(t− 1) + rand · (C(t− 1)−Xi(t− 1)) (14)

where Np denotes the number of pigeons, C denotes the centerposition, and rand represents a random number within [0, 1].

Based on (11), (12), (13) and (14), the two operators areconducted in turn to ensure the regular operation of the PIO

2620 IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 24, NO. 6, DECEMBER 2019

Fig. 6. Schematic diagram of EGPIO in detail.

algorithm. However, the number of iterations of each operatorneeds to be artificially set according to different problems,which is time-consuming. In addition, the update strategies forindividuals in PIO employ the experience of the whole flock.To a certain extent, this fixed mechanism limits the flexibilityof the algorithm, making the results fall into local optima moreeasily. Thus, to improve both the accuracy and efficiency of PIO,the intelligence level of individuals and the adaptability of thealgorithm need to be enhanced together.

B. Principle of the Hybrid Mechanism

In EGPIO, EGT is introduced to improve the efficiency andaccuracy of PIO since this theory concentrates on the behavior oflarge populations of agents who repeatedly engage in strategicinteractions [39].

To illustrate the mechanism of the proposed method, a detailedschematic diagram of EGPIO is shown in Fig. 6. In the hybridgame approach, an analogy can be made as follows:

� Each pigeon is mapped as a player in an evolutionarygame.

� Two operators are regarded as two avail-able strategies S1 and S2 with a state-spaceK={ym :

∑ym=1,m=1, 2, ym ≥ 0}, where ym ∈ R

denotes the proportion of strategy m in the population.� The average behavior obtained by following a specific

strategy constitutes the payoff matrix A.Subsequently, the game proceeds under the mechanism of

fundamental evolutionary dynamics proposed by Taylor andJonker [40], which is written as

y�m = ym(amy − yTAy) (15)

where am denotes the mth row of the payoff matrix A, andA ∈ �m×m holds all the fitness information of the populationwhich is the integration result from each individual using a single

strategy. Thus, we can define A as

A =

(a(s1)

a(s2)+a(s1)2

a(s1)+a(s2)2 a(s2)

)

(16)

wherea(sm),m ∈ {1, 2} represents the payoff of a pigeon usingstrategy m. As an ordinary differential equation expressing thedifference between the fitness of a strategy and the averagefitness in the population, (15) describes the evolution of strategyfrequencies ym. Once (16) is created, (15) is executed to generatethe associated evolutionary stable strategy (ESS) which is theoutput of the game [40]. Therefore, we obtain the ESS candi-dates at the number of iterations t as P t = (Y t

m),m ∈ {1, 2} inwhich the coefficient Y t

m,m ∈ {1, 2}, the specific value of ym,represents the ratio of strategy m in the tth iteration.

Considering the iterative process, the payoff a(sm) for strat-egy m is proposed as

a(sm) =1t

t∑

j=1

Y jm · f(Xj) (17)

where 1t describes the average during the iteration, and f(Xj)

represents the cost of pigeon X in the jth iteration. Clearly,(17) reflects the average performance of the obtained gain by aspecific strategy.

In the process of solving the dynamic equation, better resultsare constantly produced to replace the previous solution, and thegame eventually converges to an ESS that cannot be invaded byany mutant strategy [40].

C. Mathematical Expressions of the Novel Algorithm

According to the definition in [40], the values of an ESS canbe explained by the fact that they are more successfully rewardedby high reproductive rates. In our design, EGPIO is designed bymultiplying a pair of solutions Y t

1 and Y t2 obtained by (15) with

the corresponding operators. Therefore, the new position andvelocity of each pigeon are updated as

N(t) = N(t− 1)−Ndec (18)

Xi(t) = Xi(t− 1) + Vi(t) (19)

Vi(t) = Vi(t− 1) · e−Rt+rand · tr·Y t1 ·(Xgbest−Xi(t− 1))

+ rand · tr · Y t2 · (C(t)−Xi(t− 1)) (20)

whereNdec is the number of discarded pigeons in each iteration,R is the map and compass factor, tr represents the transitionfactor, and the range of a random number rand is [0, 1]. As theiteration proceeds, the roles ofXgbest andC(t) jointly determinethe position and velocity of a pigeon under Y t

1 and Y t2 satisfying

Y t1 + Y t

2 ≈ 1.As shown in (18)–(20), two independent operators are inte-

grated into a whole, which eliminates the need to set the numberof iterations one after another. Moreover, the proportion of eachoperator in PIO is dynamically determined by the solutionscalculated by (15), and the ESS ensures that the optimal resultscan eventually be obtained.

HAI et al.: MOBILE ROBOT ADRC WITH AN AUTOMATIC PARAMETER TUNING MECHANISM 2621

Fig. 7. Schematic diagram of the EGPIO-ADRC controller.

V. IMPROVED ACTIVE DISTURBANCE REJECTION CONTROL

APPROACH FOR DIRGO

In this section, a modified ADRC method employing EGPIOto automatically tune the parameters of the controller is pro-posed, named EGPIO-ADRC.

A. Structure of the Proposed Scheme

To illustrate the novel controller, a schematic diagram ofEGPIO-ADRC for the system is shown in Fig. 7. Actually, theEGPIO algorithm is adopted to automatically handle the tunableparameters, including r0, r, and h in the TD; α, β01, and β02 inESOs, and c in NLSEF. Consequently, seven key optimizationparameters for each ADRC dominating the systems in differentlocations can be viewed as seven variables in a specific functionC(r0, r, h, α, β01, β02, c) representing the control system. Dueto the identity of the model and the effect of the two separatesystems, the results of EGPIO will be input into each controllersynchronously. Therefore, the estimation of the system statesand disturbances is closer to the real values, and the performanceof the controller will be promoted.

B. Establishment of the Objective Function

Generally, two indices measure the control performance of thesystem: the dynamic performance includes angle overshoot, risetime, and settling time; the stable performance depends mainlyon stable error. In the motion system of Dirgo, the desired systemperformance is set as follows: the angle overshoot σ is smallerthan 1.5%, the rise time tr is shorter than 3 s, the settling timets is shorter than 5 s, and the stable error eSS is smaller than0.01%. Thus, the objective function f can be composed of theweighted sum of the performance indices and is given by

f = ω1 · σ + ω2 · tr + ω3 · ts + ω4 · eSS (21)

where ω1, ω2, ω3, and ω4 are weight values set to ω1 = 200,ω2 = 100,ω3 = 0.5, andω4 = 30. By minimizing the above ob-jective function, the ideal control performance can be achieved.

C. Implementation of the EGPIO Algorithm for ADRC

In each iteration process, each pigeon holding two availablestrategies acts as a participant in the evolutionary game. Guidedby the objective function, all pigeons will fly toward the optimalposition determined by the computation of the preceding dy-namic equation. In the experiments, the detailed implementation

Fig. 8. Flowchart of EGPIO employed by the ADRC optimizationproblem.

procedure of EGPIO computing the optimization parameters[r0, r, h, α, β01, β02, c] can be described as follows:

Step 1: Initialize the parameters in EGPIO, including thenumber of pigeons NP , the maximum number ofiterations Tmax, and the state of each pigeon, i.e., thepositionXi and the velocityVi in seven-dimensionalsearch space.

Step 2: Drive the motion system model in Dirgo under thecontrol of the parameters from Step 1 and calculatethe objective function according to (21). Comparethe results and find the current best values.

Step 3: Execute the iteration by following the map and com-pass operator in (11) and (12) and the landmarkoperator in (13) and (14) independently.

Step 4: Compute the payoff of each pigeon using differentstrategies by (17). The payoffs are used to composea payoff matrix in (16).

Step 5: Obtain the ESS by (15), and a pair of solutions willbe the key parameters for the execution of EGPIO.

Step 6: Execute the iterations of EGPIO by using the ESScomputed in Step 5.

Step 7: Check the stop criterion. If the convergence condi-tion is satisfied, output the optimal parameters andstop the algorithm. Otherwise, go to Step 3.

The above steps are summarized in a flowchart in Fig. 8.

VI. EXPERIMENTAL SETUP AND RESULTS

The purpose of controlling the robot is to keep it working inan unknown environment with satisfactory performance, i.e., theangle of inclination can be quickly and steadily restored to 0 rad.In addition, the failure mode and effects analysis of the motion

2622 IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 24, NO. 6, DECEMBER 2019

TABLE IIUNITS FOR MAGNETIC PROPERTIES

system in [38] indicates that disabling of the push rod is themain form of failure. Therefore, we consider the performanceof EGPIO-ADRC in both normal and fault operations.

Experiments were set up in a MATLAB/Simulink modelusing EGPIO-ADRC. The motion system model is built inSimulink. Each ADRC controller is written by an S-function,and the EGPIO algorithm is written in an M-file. The operatingenvironment of the simulation experiments in this article is aCore i7 CPU, Windows 10 operating system, and a dual-coreIntel TM. The physical memory is 8 GB, and the speed of theprocessor is 2.2 GHz. The algorithm is run on MATLAB R2017bsoftware.

A. Verification of EGPIO-ADRC

To verify the effectiveness of EGPIO-ADRC, PIO, CPIO [35],[41], and PSO [42], optimization parameters under the sameconditions are utilized for comparison. The parameters of theselected algorithms are given in Table II.

Fig. 9 shows the comparative results among PIO-ADRC,CPIO-ADRC, PSO-ADRC, and EGPIO-ADRC for a modelof push rod deformation mechanisms with different input stepvalues of 0.2, 0.4, 0.6, and 0.8, imitating various changes inangle at t = 0 s.

Apparently, the output responses show that EGPIO-ADRCsuccessfully drives the overall angle outputs of the system toreference values in the presence of disturbances. As shownin Fig. 9(a), there is no significant difference in all kinds ofindices among the selected controllers except for PIO-ADRC.Clearly, the performance of EGPIO-ADRC is prominent, i.e.,it produces a smaller steady-state error. As shown in Fig. 9(b),EGPIO performs the best for the settling time index, but theovershoot is not good enough. However, as the complexityincreases, i.e., when the disturbance value is increased to 0.6and 0.8, the advantages of EGPIO-ADRC gradually becomeobvious, as shown in Fig. 9(c) and (d). First, the superiority forthe settling time is still preserved in these two cases. Second, theoutput value converges to a position closer to the target value.Finally, the stable error becomes lower than that of the othercontrollers.

From the above results, we can conclude that the newly de-veloped control method achieves the ideal control performance

Fig. 9. Output responses of the selected controllers when the systemis in normal operation. (a) Step value is 0.2. (b) Step value is 0.4.(c) Step value is 0.6. (d) Step value is 0.8.

in our test model, including accuracy, stability, rapidity, androbustness against the disturbance. The slight disadvantage ofEGPIO-ADRC compared with CPIO-ADRC in the first casecan be neglected since the contribution of the EGT mechanismis reflected mainly in complicated problems. To further showthe superiority of EGPIO-ADRC, simulation results under cir-cumstances with complex disturbances are investigated in thefollowing section.

HAI et al.: MOBILE ROBOT ADRC WITH AN AUTOMATIC PARAMETER TUNING MECHANISM 2623

Fig. 10. (a) Values of the pair of ESSs changing over iterations for a0.2 step signal test case. (b) Average fitness evolution of the selectedmethods for a 0.2 step signal test case.

B. Convergence Analysis of EGPIO

To verify the advantages of our proposed EGPIO, curves ofthe average fitness value of 50 trials during runs for the selectedalgorithms optimizing the objective function in different initialstep signals (0.2, 0.4, 0.6, and 0.8) are presented. Further, toexplain the contribution of EGT to PIO and CPIO, the evolutionratio of each strategy in basic PIO is also investigated.

As shown in Fig. 10(a), the ESS values change, and moreremarkably, the two values converge to stable values in the fifthiteration at [0.622, 0.401]. In our design, the initial values ofthe ESS are set to [0.5, 0.5], which means that the proportionof pigeons using each strategy is equal initially. The ESS valuescorresponding to each iteration are the main coefficients in (20).In particular, equilibrium values are usually used as the bestallocation values adopted by EGPIO. Fig. 10(b) shows the aver-age fitness evolution of the selected methods. Clearly, EGPIO ismore competitive than PIO, CPIO, and PSO. The curve changeof PIO reflects its characteristics; that is, the result relies largelyon the independent search ability of the two operators. AlthoughCPIO overcomes the shortcomings of PIO, it is unreliable insearching for global optima. Interestingly, the trends of CPIOand EGPIO are generally consistent, but the dynamic adaptivemechanism brought by EGT strengthens the global search abilityof EGPIO. In Fig. 11, EGPIO also performs the best among theselected algorithms and obvious progress has been made by PIO.The result illustrates that EGPIO maintains stable performancebut, on the other hand, implies that there is a certain degree ofrandomness in standard PIO. Compared with the last case, bothPSO and CPIO fall into local optima and perform better than

Fig. 11. Average fitness evolution of the selected methods for a0.4 step signal test case.

Fig. 12. (a) Values of the pair of ESSs changing over iterations for a0.6 step signal test case. (b) Average fitness evolution of the selectedmethods for a 0.6 step signal test case.

PIO, CPIO, and social-class PIO (SCPIO). The equilibrium ESSvalues in this case are [0.629, 0.398].

Fig. 12(a) gives the ratio of the two strategies over iterationsfor a disturbance with a 0.6 rad condition. As shown, theequilibrium point converges to [0.569, 0.451] after the fifth iter-ation. A possible reason for a slight float during the convergencemay be that the pigeons try to discard the current state and searchfor better results. The evolution speeds of both CPIO and EGPIOshown in Fig. 12(b) are slower than those of PIO and PSO. PIOachieves outstanding results in this case, though not as much asEGPIO in global optimization.

The comparative results for a disturbance with 0.8 rad arepresented in Fig. 13, representing the most difficult conditionsince Dirgo needs to be restored to the original state underthe maximum angle interference. Nevertheless, all algorithms

2624 IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 24, NO. 6, DECEMBER 2019

Fig. 13. Average fitness evolution of the selected methods for a0.8 step signal test case.

have made remarkable progress in accuracy. Although theconvergence speed of EGPIO is not better than that of the others,it still reaches the global optima among them. In addition, thedisparity is within the scope of our acceptance.

Based on the important information above illustrating thepractical optimization process, it can be concluded that favorableperformance is achieved by EGPIO and that the mechanisms ofEGT have a positive effect on promoting the global optimizationcapability. From the results, we can confirm that the selectedalgorithms are sufficiently competitive so that EGPIO has astronger ability to perform a precise search.

C. Evaluation of the Fault-Tolerant Capability

Finally, we considered the fault-tolerant capability of thesecontrollers. The effects of faults were analyzed in [38]. In thispart, three failure states are chosen due to stagnation of the pushrod, which is the most serious negative effect on the system.The test environment is operated under the condition that thestep signal of the system is 0.4 rad at 0 s.

1) Case A: Stagnation Occurs With 0.4 Rad at t = 0 s: Atthis moment, the desired position of the push rods is level, thatis, the reference angle is 0 rad. First, we consider the conditionof the push rods in the front of Dirgo if the push rods arestuck. As shown in Fig. 14(a), the performance of PSO-ADRC isexcellent, i.e., it returns to the reference state in a very short time,and almost no continuous fluctuation exists. By comparison,EGPIO-ADRC does not behave as well as expected, althoughit is superior to the other two controllers. In particular, EGPIO-ADRC has a similar trend of variation with CPIO-ADRC buthas smaller overshoots and shorter time. As shown in Fig. 14(b),PIO-ADRC still provides the worst resilience performance forthe system in the case of the presence of disturbances and failureswhen both the front and back push rods are stuck. In contrastto the previous result, the robustness property of PSO-ADRCagainst more complex uncertainties is unsatisfactory. Due to thebenefits of the adaptive component in the design, EGPIO-ADRCprovides a lower steady-state error and faster transient response.

2) Case B: Stagnation Occurs With 0.4 Rad at t = 10 s:Clearly, case B is more complicated than the previous case,which can be seen from the curves of recurrent oscillation for thetest controllers in Fig. 15. Despite the convergence properties ofthe selected methods, the detailed indices are not exactly the

Fig. 14. Output responses of the selected controllers when the systemis in fault operation with a type A failure. (a) The front push rods arestuck. (b) Both the front and back push rods are stuck.

Fig. 15. Output responses of the selected controllers when the systemis in fault operation with a type B failure: stagnation occurs with 0.4 radat t = 10 s.

same: PSO-ADRC provides very fast transient response andlow tracking error and shows ideal performance; EGPIO-ADRCprovides superior performance compared to PIO-ADRC andCPIO-ADRC. An advantage of EGPIO over standard PIO andCPIO is that the adaptive mechanism offered by EGT expandsthe choice space for the population. However, PIO and its twovariants appear deficient by comparison.

3) Case C: Stagnation Occurs With x Rad at t = 10 s: Incase C, a stuck situation appears at the last task completiontime, and angle x is the current value. Based on this condition, a0.4 rad step signal is given at that moment. The system responseis shown in Fig. 16. In the presence of the most practical andcomplex fault, EGPIO-ADRC provides the best performancesince it possesses low steady-state error, small overshoot, andshort settling time; the figure illustrates that both EGPIO-ADRC

HAI et al.: MOBILE ROBOT ADRC WITH AN AUTOMATIC PARAMETER TUNING MECHANISM 2625

Fig. 16. Output responses of the selected controllers when the systemis in fault operation with a type C failure: stagnation occurs at the lasttask completion time.

and CPIO-ADRC are capable of fault tolerance. In contrast to theprevious case, PSO-ADRC is the worst controller in this case.For this reason, it is inferior to our proposed EGPIO-ADRC.

VII. CONCLUSION

A novel EGPIO-ADRC control approach has been investi-gated to realize fault-tolerant control for a self-developed mobilerobot. In the proposed method, EGPIO is utilized as an automaticparameter tuning mechanism for an ADRC controller. An ad-vantage of EGPIO-ADRC is that it can improve the efficiency ofa basic ADRC controller by applying the EGPIO algorithm andyielding the optimal strategy for different problems. To improvethe efficiency of standard PIO, EGT is introduced to producean ESS for individuals during the search process and providethe optimization with adaptability. This creative combinationcontributes to convergence efficiency compared with that ofPIO, CPIO, and PSO. Based on the feedback information fromthe motion system of the robot, the parameters in ADRC canbe optimized by the adaptive dynamic procedure in EGPIO.Therefore, a convenient parameter selection scheme can beobtained that not only makes the results more persuasive butalso improves the efficiency of the control method. Simulationresults verify the superior performance of the proposed methodin terms of disturbance rejection in normal operation and faultcompensation in fault operation.

In future work, theoretical developments including stabilityanalysis of the overall closed-loop control scheme based on theLyapunov method, precise kinematic constraints of the system,and a convergence proof of the proposed algorithm will be fur-ther discussed. We also expect that the proposed control schemecould be implemented in another type of function where controlenergy is included. Further, more complicated fault-patterns willbe considered and our proposed EGPIO will be compared withmore up-to-date algorithms and developed in other optimizationaspects.

REFERENCES

[1] J. Zhou, X. L. Ding, and Y. Y. Qing, “Automatic planning and coordinatedcontrol for redundant dual-arm space robot system,” Ind. Robot, vol. 38,no. 1, pp. 27–37, 2011.

[2] L. Wang, T. Chai, and C. Yang, “Neural-network-based contouring controlfor robotic manipulators in operational space,” IEEE Trans. Control Syst.Technol., vol. 20, no. 4, pp. 1073–1080, Jul. 2012.

[3] J. C. L. Barreto et al., “Design and implementation of model-predictive control with friction compensation on an omnidirectional mo-bile robot,” IEEE/ASME Trans. Mechatronics, vol. 19, no. 2, pp. 467–476,Apr. 2014.

[4] D. Guo, X. Feng, and L. Yan, “New pseudoinverse-based path-planningscheme with PID characteristic for redundant robot manipulators in thepresence of noise,” IEEE Trans. Control Syst. Technol., vol. 26, no. 6,pp. 2008–2019, Nov. 2018.

[5] Q. Meng et al., “Adaptive sliding mode fault-tolerant control of theuncertain Stewart platform based on offline multibody dynamics,”IEEE/ASME Trans. Mechatronics, vol. 19, no. 3, pp. 882–894, Jun.2014.

[6] K. Haninger and M. Tomizuka, “Robust passivity and passivity relaxationfor impedance control of flexible-joint robots with inner-loop torquecontrol,” IEEE/ASME Trans. Mechatronics, vol. 23, no. 6, pp. 2671–2680,Dec. 2017.

[7] M. Jin et al., “Robust control of robot manipulators using inclusive andenhanced time delay control,” IEEE/ASME Trans. Mechatronics, vol. 22,no. 5, pp. 2141–2152, Oct. 2017.

[8] G. Rigatos, P. Siano, and G. Raffo, “A nonlinear H-infinity control methodfor multi-DOF robotic manipulators,” Nonlinear Dyn., vol. 88, no. 1,pp. 329–348, 2017.

[9] M. Makarov et al., “Modeling and preview H� control design for mo-tion control of elastic-joint robots with uncertainties,” IEEE Trans. Ind.Electron., vol. 63, no. 10, pp. 6429–6438, Oct. 2016.

[10] L. H. Wang, “Adaptive control of robot manipulators with uncertainkinematics and dynamics,” IEEE Trans. Autom. Control, vol. 62, no. 2,pp. 948–954, Feb. 2017.

[11] B. Brahmi et al., “Adaptive tracking control of an exoskeleton robotwith uncertain dynamics based on estimated time-delay control,”IEEE/ASME Trans. Mechatronics, vol. 23, no. 2, pp. 575–585, Apr.2018.

[12] M. Van, M. Mavrovouniotis, and S. Ge, “An adaptive backstepping non-singular fast terminal sliding mode control for robust fault tolerant controlof robot manipulators,” IEEE Trans. Syst., Man, Cybern., vol. 49, no. 7,pp. 1448–1458, Jul. 2019.

[13] J. Mukherjee, S. Mukherjee, and I. N. Kar, “Sliding mode control of planarsnake robot with uncertainty using virtual holonomic constraints,” IEEERobot. Autom. Lett, vol. 2, no. 2, pp. 1077–1084, 2017.

[14] N. Martínez-Fonseca et al., “Robust disturbance rejection control of abiped robotic system using high-order extended state observer,” ISA Trans.,vol. 62, pp. 276–286, May 2016.

[15] B. Ahi and A. Nobakhti, “Hardware implementation of an ADRC con-troller on a gimbal mechanism,” IEEE Trans. Control Syst. Technol.,vol. 26, no. 6, pp. 2268–2275, Nov. 2018.

[16] G. Calafiore and M. C. Campi, “The scenario approach to robust con-trol design,” IEEE Trans. Autom. Control, vol. 51, no. 5, pp. 742–753,May 2006.

[17] G. Tao, “Multivariable adaptive control: A survey,” Automatica, vol. 50,no. 11, pp. 2737–2764, 2014.

[18] V. I. Utkin, “Sliding mode control design principles and applications toelectric drives,” IEEE Trans. Ind. Electron., vol. 40, no. 1, pp. 23–36,Feb. 1993.

[19] J. Han, “Auto-disturbances-rejection controller and its application,” Con-trol Decis., vol. 13, no. 1, pp. 19–23, 1998.

[20] J. Han, “From PID to active disturbance rejection control,” IEEE Trans.Ind. Electron., vol. 56, no. 3, pp. 900–906, Mar. 2009.

[21] Z. Gao, “Active disturbance rejection control: A paradigm shift infeedback control system design,” in Proc. Amer. Control Conf., IEEE,2006.

[22] B. Ahi and M. Haeri, “Linear active disturbance rejection control fromthe practical aspects,” IEEE/ASME Trans. Mechatronics, vol. 23, no. 6,pp. 2909–2919, Dec. 2018.

[23] X. H. Qi et al., “On the robust stability of active disturbance rejectioncontrol for SISO systems,” Circuits Syst. Signal Process., vol. 36, no. 1,pp. 65–81, 2017.

[24] C. Aguilar-Ibanez, H. Sira-Ramirez, and J. A. Acosta, “Stability of ac-tive disturbance rejection control for uncertain systems: A Lyapunovperspective,” Int. J. Robust Nonlinear Control, vol. 27, pp. 4541–4553,2017.

[25] Y. Huang and W. C. Xue, “Active disturbance rejection control: Method-ology and theoretical analysis,” ISA Trans., vol. 53, no. 4. pp. 963–976,Jul. 2014.

[26] Z. Chu, Y. Sun, C. Wu, and N. Sepehri, “Active disturbance re-jection control applied to automated steering for lane keeping inautonomous vehicles,” Control Eng. Pract., vol. 74, pp. 13–21,2018.

2626 IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 24, NO. 6, DECEMBER 2019

[27] M. Baquero-Suárez et al., “A robust two-stage active disturbance rejectioncontrol for the stabilization of a riderless bicycle,” Multibody Syst. Dyn.,to be published., doi: 10.1007/s11044-018-9614-y.

[28] H. B. Duan and P. X. Qiao, “Pigeon-inspired optimization: A new swarmintelligence optimizer for air robot path planning,” Int. J. Intell. Comput.Cybern., vol. 7, no. 1, pp. 24–37, 2014.

[29] Y. M. Deng and H. B. Duan, “Control parameter design for automaticcarrier landing system via pigeon-inspired optimization,” Nonlinear Dyn.,vol. 85, pp. 1–10, 2016.

[30] R. Dou and H. B. Duan, “Levy flight based pigeon-inspired optimizationfor control parameters optimization in automatic carrier landing system,”Aerosp. Sci. Technol., vol. 61, pp. 11–20, 2017.

[31] H. B. Duan and X. H. Wang, “Echo state networks with orthogonal pigeon-inspired optimization for image restoration,” IEEE Trans. Neural Netw.Learn. Syst., vol. 27, no. 11, pp. 2413–2425, 2016.

[32] D. F. Zhang, H. B. Duan, and Y. J. Yang, “Active disturbance rejection con-trol for small unmanned helicopters via levy flight-based pigeon-inspiredoptimization,” Aircr. Eng. Aerosp. Technol., vol. 89, no. 6, pp. 946–952,2017.

[33] H. B. Duan, H. X. Qiu, and Y. M. Fan, “Unmanned aerial vehicle for-mation cooperative control based on predatory escaping pigeon inspiredoptimization,” Scientia Sinica, vol. 45, no. 6, pp. 559–572, 2015.

[34] J. M. Smith, Evolution and the Theory of Games. Cambridge, U.K.:Cambridge University Press, 1982, pp. 41–45.

[35] X. S. Hai et al., “A novel adaptive pigeon-inspired optimization algorithmbased on evolutionary game theory,” Sci. China Info. Sci., to be published,doi: org/10.1007/s11432-018-9923-6.

[36] C. Samson, P. Morin, and R. Lenain, “Modeling and control of wheeledmobile robots,” in Springer Handbook of Robotics, 2nd ed., 2016,pp. 1235–1263.

[37] S. Nolfi, J. Bongard, P. Husbands, and D. Floreano, “Evolutionaryrobotics,” in Springer Handbook of Robotics, 2nd ed., 2016, pp. 2035–2067.

[38] J. J. Cui et al., “On the robustness and reliability in the pose deforma-tion system of mobile robots,” IEEE Access, vol. 6, pp. 29747–29756,May 2018, doi: 10.1109/ACCESS.2018.2835836.

[39] C. Leboucher et al., “A swarm intelligence method combined to evolu-tionary game theory applied to the resources allocation problem,” Int. J.Swarm Intell. Res., vol. 3, no. 2, pp. 20–38, 2012

[40] P. Taylor and L. Jonker, “Evolutionary stable strategies and game dynam-ics,” Math. Bios, vol. 40, pp. 145–156, 1978.

[41] H. B. Duan, H. X. Qiu, and Y. M. Fan, “Unmanned aerial vehicle for-mation cooperative control based on predatory escaping pigeon-inspiredoptimization,” Scientia Sinica Technologica, vol. 45, pp. 559–572, 2015.

[42] Y. H. Shi and R. C. Eberhart, “A modified particle swarm optimizer,” inProc. IEEE World Congr. Comput. Intell., 1998, pp. 69–71.

Xingshuo Hai was born in Beijing, China, in1993. He recieved the B.S. degree in naviga-tion guidance and control from the School ofAutomation Science and Electrical Engineering,Beihang University (BUAA), Beijing, China, in2015. Currently, he is working toward the Ph.D.degree in system engineering with the Schoolof Reliability and Systems Engineering, BeihangUniversity (BUAA).

His research interests include systems en-gineering, fault-tolerant control, and intelligent

optimization algorithms.

Zili Wang received the B.S. degree in automaticcontrol from Beihang University (BUAA), Beijing,China, in 1985, and the M.S. degree in reliabilityand systems engineering from the School ofReliability and Systems Engineering of BeihangUniversity, Beijing, in 1988.

He is currently a Professor and Presidentwith the School of Reliability and Systems Engi-neering, Beihang University, Beijing, China. Heis also a Supervisor of doctoral and master’sstudents. His current research interests include

systems engineering, quality engineering, and prognostics and healthmanagement.

Qiang Feng (M’13) received the B.S. degree inmechanical engineering and the Ph.D. degree inreliability engineering and systems engineeringfrom Beihang University, Beijing, China in 2001and 2009, respectively.

He is currently an Assistant Professor insystem engineering and a member with theFaculty of Systems Engineering, School of Reli-ability and Systems Engineering, Beihang Uni-versity, Beijing, China. He is a Supervisor ofM.S. students and teaches two courses for un-

dergraduate and master’s students. His current research interests in-clude reliability modeling of complex systems, maintenance planning,and integrated design of product reliability and performance.

Yi Ren received the Ph.D. degree in reliabilityengineering and systems engineering from Bei-hang University, Beijing, China, in 2012.

He is currently a Professor in system engi-neering, a Reliability Specialist, and a memberwith the Faculty of System Engineering, De-partment of System Engineering of EngineeringTechnology, Beihang University, Beijing, China.He has over ten years of research and teachingexperience in reliability engineering and systemengineering. He is the Team Leader of the KW-

GARMS reliability engineering software platform. His recent researchinterests include the reliability of electronics, model-based reliabilitysystem engineering (MBRSE), computer-aided reliability maintainabilityand supportability design, and product lifecycle management.

Dr. Yen holds six Chinese ministry-level professional awards and oneChinese national-level professional award.

Binghui Xu received the B.S. degree in reliabil-ity and system safety from Beihang University,Beijing, China, in 2018, where he is currentlyworking toward the M.S. degree in system en-gineering.

His research interests include reliability as-sessment of complex systems and prognosticsand health management (PHM).

Jingjing Cui received the B.S. degree in controlscience and engineering from Zhejiang Univer-sity, Zhejiang, China, in 2013. She is currentlyworking toward the Ph.D. degree in reliabilityand systems engineering at the School of Re-liability and Systems Engineering, Beihang Uni-versity, Beijing, China.

Her research interests include system reliabil-ity modeling, active disturbance rejection controltechniques, and system reliability analysis.

Haibin Duan (M’07–SM’08) received the Ph.D.degree in control theory and engineering fromthe Nanjing University of Aeronautics and As-tronautics (NUAA), Nanjing, China, in 2005.

He was an Academic Visitor with the NationalUniversity of Singapore (NUS), Singapore, in2007, a Senior Visiting Scholar with the Univer-sity of Suwon (USW) of South Korea in 2011,an Engineer with the Shenyang Aircraft DesignResearch Institute, Aviation Industry of China(AVIC) in 2006, and a Technician with the China

Aviation Motor Control System Institute from 1996 to 2000. He is cur-rently a Full Professor in navigation guidance and control with the Schoolof Automation Science and Electrical Engineering, Beihang University,Beijing, China. His current research interests include bio-inspired com-puting, biological computer vision, and multi-UAV autonomous formationcontrol.