Research Article An Adaptive HH -Based Formation Control for Multirobot Systems

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1 ISRN Robotics Volume 3, Article ID 9487, pages Research Article An Adaptive HH -Based Formation Control for Multirobot Systems Faridoon Shabani, Bijan Ranjbar, and Ali Ghadamyari School of Electrical Engineering, Shiraz University, Shiraz, Iran Correspondence should be addressed to Faridoon Shabani; Received 7 April ; Accepted 3 May Academic Editors: J. B. Koeneman and G. S. Virk Copyright 3 Faridoon Shabani et al. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We describe a decentralized formation problem for multiple robots, where an HH formation controller is proposed. e network of dynamic agents with external disturbances and uncertainties are discussed in formation problems. We rst describe how to design social potential elds to obtain a formation with the shape of a polygon. en, we provide a formal proof of the asymptotic stability of the system, based on the de nition of a proper Lyapunov function and HH technique. e advantages of the proposed controller can be listed as robustness to input nonlinearity, external disturbances, and model uncertainties, while applicability on a group of any autonomous systems with nn-degrees of freedom. Finally, simulation results are demonstrated for a multiagent formation problem of a group of six robots, illustrating the effective attenuation of approximation error and external disturbances, even in the case of agent failure or leader tracking.. Introduction All around the world, nature presents examples of collective behavior in groups of insects, birds, and shes. is behavior has produced sophisticated functions of the group that cannot be achieved by individual members [, ]. erefore, the research on the coordination of robotic swarms has attracted considerable attention. Taking the advantages of distributed sensing and actuation, a robotic swarm can perform some cooperative tasks such as moving a large object that is usually not executable by a single robot [3 7]. Applications about the analysis and design of robotic swarms included autonomous unmanned aerial vehicles, congestion control of communication networks, and distributed sensor networks autonomous, and so forth [,, 8 ]. In general, a robotic formation problem is de ned as the organization of a swarm of agents into a particular shape in a D or 3D space [8]. is kind of control strategy can be applied into several different elds. For example, in the industrial eld, this formation control strategy can be applied to a group of Automated Guided Vehicles (AGVs) moving in a warehouse for goods delivery. e main idea is to make a group of AGVs cooperatively deliver a certain amount of goods, moving in a formation. e creation of a formation with the desired shape is useful to precisely constrain the action zone of the AGVs, thus reducing the chance of collisions with other entities (e.g., human guided vehicles). In the literature, many different approaches to formation control can be found. e main existing approaches can be divided into two categories: centralized [] and distributed []. Because of the intrinsic unreliability of centralized methods, we focus our attention to distributed ones: all the agents are equal, and if one of them stops working, the other ones can still complete their task. Several formation control strategies can be found as potential elds [8], behaviorbased [3], leader-following [4 6], graph-theoretic [], and virtual structure approaches [7, 8]. In recent years, some methods based on potential elds are integrated with some nonlinear control schemes such as feedback linearization method (e.g., Sliding Mode Control (SMC)), which concludes in more robust formation control designs of dynamic agents [5 8]. For example, Takahashi et al. [5] proposed an SMC-based formation control scheme for multiple mobile robots, using the leaderfollowing strategy, in which they de ned some performance

2 ISRN Robotics indexes, so that robots can be controlled according to their ability. Defoort et al. [6] also developed a robust coordinated control scheme based on leader-follower approach to achieve formation maneuvers. ey used rst- and second-order SMC to address the formation problem of mobile robots of unicycle type with two driving wheels. Moreover, Cheaha et al. [7] presented a region-based shape controller for a swarm of fully actuated robots, where a linear approximator was used to approximate the unknown dynamic model and an SMC controller integrated with arti cial potential functions was used to satisfy a predetermined geometric D formation. Recently, HH optimal control techniques have been found to be an effective solution to treat robust stabilization and tracking problems, in presence of external disturbances and system uncertainties [9 4]. In an HH control technique, the main design goal is to force the gain from unmodelled dynamics, external disturbances, and approximation errors to be equal or less than a prescribed disturbance attenuation level (HH attenuation constraint) [9]. is goal is generally represented as a Linear Matrix Inequality (LMI) problem. In the traditional HH control the exact model of the system must be known. However, in order to propose a robust control method, an integration between this robust scheme with fuzzy logic approximators can propose effective controllers for uncertain dynamic models [5]. Since Zadeh [6] initiated the fuzzy set theory, fuzzy logic systems (FLS) have been widely applied to many real world applications [7 3]. However, fuzzy control has not been viewed as a rigorous science due to a lack of formal synthesis techniques which guarantee the very basic requirements of global stability and acceptable performance. In fact, if the mathematical model of a robot is known, then conventional linear and nonlinear approximation methods should be given higher priority. However, fuzzy control should be useful in situations where () there is no acceptable mathematical model for the robot and () there are experienced human operators who can satisfactorily approximate the plant and provide qualitative control rules in terms of vague and fuzzy sentences. ere are many practical situations where both () and () are true. Besides, FLS schemes have been widely used in motion control of single robots [3, 3]. Using FLS integrated with HH control technique can improve the robustness of controller and ensures the stability [5]. In this paper, a geometric formation is considered as the goal and an arti cial potential is de ned to guide the agents through this formation. A partially unknown nonlinear dynamic model is adopted to each n-degrees of freedom agent. erefore, an adaptive interval type- fuzzy approximator is combined with HH control technique to propose a novel decentralized adaptive fuzzy formation control methodology, with robust characteristics. e main advantage of this control strategy is insensitivity to robot dynamic uncertainties, external disturbances, and input nonlinearities. Moreover, in existing adaptive nonlinear control methodologies which are based on SMC control (e.g., [7]), each agent approximator needs to know the position and velocity of all other robots to approximate the unknown model dynamics; however, in the current proposed decentralized strategy, only the position and velocity of each robot are enough to be known to its approximator. e rest of this paper is organized as follows: Section presents the system description, problem formulation, and potential function evaluation. An introduction of interval type- fuzzy logics systems is described in Section 3. Design of the proposed controller and stability analysis are discussed in Sections 4 and 5, respectively. Simulation results are included in Section 6 and Section 7 provides the concluding remarks.. System Description and Problem Formulation e major goal in this study is to solve a multiagent formation control problem (i.e., controlling the relative position and orientation of the agents to create a desirable formation). One of the effective solutions for this problem is using an electrostatic-like potential function design which guides the agents through continues smooth paths and avoids agent collisions. Such a potential function design has been discussed in various papers (e.g., [,, 8, 8]). erefore, in Section. we will explain a simple potential function design, in order to solve the formation control of a group of point massless agents, where the kinematic of the ith agent is considered as zz ii uu ii, ii i iii ii i i ii}, () in which zz ii RR nn is the coordinate matrix (for a robot with n- degrees of freedom) and uu ii RR nn denotes the control inputs. However, one of the main shortcomings of this kinematic model is that it does not correspond to the dynamics of realistic agents. To overcome this shortcoming more general dynamic models like unicycle models [33] or other wheeled vehicle models can be discussed. In Section. one of the most general n-degrees of freedom dynamic models of real robots is considered to propose more realistic solutions for formation control of multiagent systems. e main feature of this model is that any agent (robot) with nn-degrees of freedom (e.g., Autonomous Underwater Vehicles (AUVs) [34], Unmanned Aerial Vehicles (UAVs) [35, 36], etc.) can be adopted to this model... Formation Control Massless Agents. To propose a control law, an arti cial potential function is designed. is potential function can be comprised of interagent interactions, environmental effects (e.g., obstacles, goals, etc.), or other exceptional terms. Consider the pairwise potential elds, which are de ned between agents as FF LL zz ii zz jj, iii ii i iii ii i i ii}, () where LL is designed to de ne a proper interagent potential function. It is assumed that each agent senses the resultant potential of all other agents.

3 ISRN Robotics 3 e overall potential function is proposed to be in the form of FF F jjjjjjj LL zz ii zz jj + QQ ii zz ii, (3) where QQ ii de nes the global potential of each agent. Finally, the following three assumptions for potential function are considered [7, 8]. Assumption. FF is continuously differentiable. Assumption. FF is strictly convex. Assumption 3. FF is positive de nite. For example, the following potential function can be chosen for a desired polygonal formation in a D Space: FF F jjjjjjj zz ii zz jj dd + zz ii rr ii. (4) At the rst step, to propose a solution for multiagent formation control, the steepest descent direction [8, 7, 8] is chosen as ff ii, (5) ii and the control law uu ii ff ii, ii i iii ii i i ii} (6) is proposed. By substituting (6) in () the kinematic model is obtained as zz ii ff ii, ii i iii ii i i ii}, (7) ii which can be rewritten in the matrix form as ZZ Z ZZZZ where ZZ Z ZZZ, zz,..., zz nn ] is the overall generalized coordinate vector. In the next subsection, it is proposed to assume the multiagent system with a general dynamic model. Furthermore, in Section 3 a robust adaptive fuzzy controller using an HH approach is used to force the satisfaction of (7). In other words the proposed controller is designed to enforce the speed of each agent along the negative gradient of potential function in (7)... Formation Control of Robots with Dynamic Models. In this subsection a general dynamic model [37] is addressed to represent any kind of autonomous n-degrees of freedom system. is model has been previously used in some existing works (e.g., [7, 8]). Consider a group of fully autonomous agents. e dynamics of the ith simple agent is strongly nonlinear [37] and can be written in the general form MM zz ii zz ii + CC zz ii, zz ii zz ii + gg zz ii uu ii, (8) where zz ii RR nn is the coordinate matrix (for a robot with n-degrees of freedom); MMMMM ii ) RR nnnnn is a symmetric positive de nite matrix and represents the inertia coe cients. CCCCC ii, zz ii ) RR nnnnn is the matrix of centripetal, Coriolis, damping, and rolling resistance forces; ggggg ii ) RR nn is an nnvector of gravitational forces and uu ii RR nn denotes the control inputs. In most practical control problems of multiagent systems the inertia matrix MMMMM ii ) is a known constant matrix independent of zz ii. erefore, the following assumption is considered. Assumption 4. MM is the inertia matrix of robots, which is assumed to be a known and constant matrix. Let us rewrite (8) as MMzz ii + CC zz ii, zz ii zz ii + gg zz ii uu ii. (9) It is straightforward to rewrite (9) as zz ii MM CC zz ii, zz ii zz ii MM gg zz ii + MM uu ii. () In the next sections, the dynamic of each single agent will be assumed to be in the form of (). 3. Interval Type- Fuzzy Logic System In this section, the interval type- fuzzy set and the inference of the type- fuzzy logic system will be presented. A type- fuzzy set in universal set XX is denoted as AA which is characterized by a type- membership function uuaa (xxx in (). e uuaa (xxx can be referred to as a secondary membership function or referred to as a secondary set, which is a type- fuzzy set in [, ]. In (3), ff xx (uuu is a secondary grade, which is the amplitude of a secondary membership function; that is, ff xx (uuu u u. e domain of a secondary membership function is called the primary membership of xx. In (3), JJ xx is the primary membership of xx, where uu u uu xx [, ] for all xx x xx; μμ is a fuzzy set in [, ], rather than a crisp point in [, ], uuaa AAA (xx) xxxxx xx xxxxx uuuuuxx ff xx (uu) /uu, JJ xx xx [, ]. () When ff xx (uuu u uu for all uu u uu xx [, ], then the secondary MFs are interval sets such that uuaa (xxx in (3) can be called an interval type- MF. erefore, the type- fuzzy set can be rewritten as uuaa AA A AA (xx) xxxxx xx xxxxx uuuuuxx /uu, JJ xx xx [, ]. () Also, a Gaussian primary MF with uncertain mean and xed standard deviation having an interval type- secondary MF can be called an interval type- Gaussian MF (3). It can be expressed as uuaa (xx) exp x xx xx σσ, mm m mm, mm. (3)

4 4 ISRN Robotics It is obvious that the type- fuzzy set is in a region, called a footprint of uncertainty (FOU) and is bounded by an upper MF and a lower MF, which are denoted as uu AA (xxx and uu AA (xxx, respectively. Hence, (3) can be reexpressed as AA A AA xxxxx uuuuu AA (xx),uu /uu AA (xx) xx. (4) A type- fuzzy logic system (FLS) is very similar to a type- FLS as shown in Figure ; the major structure difference being that the defuzzi er block of a type- FLS is replaced by the output processing block in a type- FLS which consists of type-reduction followed by defuzzi cation. ere are ve main parts in a type- FLS: fuzzi er, rule base, inference engine, type reducer, and defuzzi er. A type- FLS is a mapping ff f f pp R. A er defuzzi cation, fuzzy inference, type reduction, and defuzzi cation, a crisp output can be obtained. Consider a type- FLS having pp inputs xx XX,, xx XX pp and one output yy y yy. e type- fuzzy rule base consists of a collection of IF-THEN rules, as in the type- case. We assume there are MM rules and the rule of a type- relation between the input space XX XX XX pp and the output space YY can be expressed as RR IF xx isff ll and and xx pp isff ll pp, THEN yy isgg ll, ll l ll ll l l lll (5) whereff ll jj are antecedent type- sets (jj j jj jj j j jj) andgg ll s are consequent type- sets. e inference engine combines rules and gives a mapping from input type- fuzzy sets to output type- fuzzy sets. To achieve this process, we have to compute unions and intersection of type- sets, as well as compositions of type- relations. e output of inference engine block is a type- set. By using the extension principle of type- defuzzi cation method, type-reduction takes us from type- output sets of the FLS to a type- set called the type-reduced set. is set may then be defuzzi ed to obtain a single crisp value. ere are many kinds of type-reduction, such as centroid, height, modi ed weight, and center-of-sets. e center-ofsets type reduction will be used in this paper and can be expressed as: YY cos YY,, YY MM, FF,, FF MM yy ll, yy rr yy yy MM ff ff MM MM ffii yy ii / MM ffii, (6) where YY cos is the interval set determined by two end points yy ll and yy rr, and ff ii FF ii [ff ii, ff ii ]. In the meantime, an interval type- FLS with singleton fuzzi cation and meet under minimum or product t-norm ff ii and ff ii can be obtained as ff ii μμ xx FF ii μμ xx FF ii pp, (7) pp ff ii μμ FF ii xx μμ FF ii pp xx pp. (8) Also, yy ii YY ii and YY ii [yy ii ll, yyii rr ] are the centroid of the type- interval consequent set GG ii. For any value yy y yy cos, yy can be expressed as yy y MM ffii yy ii MM ffii, (9) where yy is a monotonic increasing function with respect to yy ii. Also, yy ll is the minimum associated only with yy ii ll, and yy rr is the maximum associated only with yy ii rr. Note that yy ll and yy rr depend only on mixture of ff ii or ff ii values. erefore, the le most point yy ll and the right-most point yy rr can be expressed as a fuzzy basis function (FBF) expansion, that is, yy ll MM ffii ll yyii ll MM ffii ll yy rr MM ffii rr yyii rr MM ffii rr MM MM yy ii ll ξξii ll, yy ii rr ξξii rr, () respectively, where ξξ ii ll ff ii ll / MM ffii and ξξ ii rr ff ii rr / MM ffii. If the FBF vector denoted as ξξ ll [ξξ ll, ξξ ll,, ξξ MM ll ] and ξξ rr [ξξ rr, ξξ rr,, ξξ MM rr ], and let yy TT ll [yy ll, yy ll,, yymm ll ] and yy TT rr [yy rr, yy rr,, yymm rr ], then () can be rewritten as yy ll MM ffii ll yyii ll MM ffii ll yy rr MM ffii rr yyii rr MM ffii rr MM MM yy ii ll ξξii ll yy TT ll ξξ ll, () yy ii rr ξξii rr yy TT rr ξξ rr. () For illustrative purposes, we brie y provide the computation procedure for yy rr. Without loss of generality, assume the yy ii rr s are arranged in ascending order, that is, yy rr yy rr yy MM rr. Step. Compute yy rr in () by initially setting ff ii rr (ffii +ff ii )/ for ii i ii ii i i ii, where ff ii and ff ii have been precomputed by (8) and (9) and let yy rr yy rr. Step. Find RR RR R RR R RR R RR such that yy RR rr yy rr yyrrrr rr. Step 3. Compute yy rr in () with ff ii rr ffii for ii i ii and ff ii rr ffii for ii i ii and let yy rr yy rr. Step 4. If yy rr yy rr, then go to Step 5, If yy rr set yy rr yy rr. Step 5. Set yy rr equal to yy rr and return to Step. yy rr, then stop and e point to separate two sides by number R can be decided from the above algorithm, one side using lower ring

5 ISRN Robotics 5 Rule base Defuzzifier Crisp Output y Input X Fuzzifier Type reducer Type Reduced set Fuzzy Input sets Inference engine Fuzzy Output sets FIGURE : e structure of the type- fuzzy logic system. strengths ff ii s and another side using upper ring strengths ff ii s. erefore, the yy rr in () can be rewritten as yy rr RR ffii yy ii rr + MM iii ffii yy ii rr RR ffii + MM iii ffii yy rr QQ rr QQ rr yy rr ξξ TT Θ rr rr, RR qq ii rr yyii rr + MM qq ii rr yyii rr iii (3) where qq ii ff ii /DD rr rr, qq ii rr ff ii /DD rr and DD rr ( RR ffii + MM iii ffii ). In the meantime, we have QQ rr [qq, rr qq,, rr qqrr], rr QQ rr [qq rr, qq rr,, qqrr rr ], ξξtt QQ QQ rr, and Θ TT rr rr rr yy yy rr rr. e procedure to compute yy ll is similar to compute yy rr. Just in Step, we determine LL LL L LL L LL L LL, such that yy LL ll yy ll yy LLLL ll and in Step 3 let ff ii ll ffii for ii i ii and ff ii ll ffii for ii i ii. en yy ll in () can also be rewritten as yy ll LL ffii yy ii ll + MM iii ffii yy ii ll LL ffii + MM iii ffii yy ll QQ ll QQ ll yy ll ξξ TT Θ ll ll, LL qq ii ll yyii ll + MM qq ii ll yyii ll iii (4) where qq ii ff ii /DD ll ll, qq ii ll ff ii /DD ll and DD ll ( LL ffii + MM iii ffii ). In the meantime, we have QQ ll [qq, ll qq,, ll qqrr], ll QQ ll [qq ll, qq ll,, qqrr ll ], ξξtt QQ ll ll QQ ll, and Θ TT ll yy ll yy ll. e defuzzi ed crisp value from an interval type- F S is obtained as 4. Controller Design Methodology In this section a novel formation error based on the integral of formation gradient (5) will be proposed. en, a robust HH controller will be designed and a fuzzy logic system will be utilized to approximate the unknown parts of dynamic models. e main feature of the proposed novel control scheme is its decentralized characteristic, robustness to external disturbances, input nonlinearities, and measurement noises. Besides, by using the proposed controller, the formation can be achieved from any initial conditions. Consider, the novel formation error for the iith robot as tt ee ii (tt) zz ii (tt) + ff ii (ττ) ddddd (6) where ee ii RR nn, zz ii represents the coordinate vector of iith robot in () and ff ii is the gradient of potential function de ned in (7). It is straightforward to write the rst and second derivatives of (6) as ee ii (tt) zz ii (tt) + ff ii, ee ii (tt) zz ii (tt) + ff ii. (7) Our design goal is to propose an adaptive fuzzy controller so that ee ii + kk ee ii + kk ee ii (8) is achieved, where kk and kk are chosen to make (8) asymptotically stable. To design the controller, consider the control law proposed as where uu ii MM HH ii zz ii, zz ii ff ii kk ee ii kk ee ii, (9) where (/) ξξ TT rr yy xx yy ll + yy rr ξξtt rr Θ rr + ξξtt ll Θ ll Θ rr ξξtt ξξ TT ξξ TT Θ, rr ll Θ ll ξξ TT ll ξξ TT and Θ TT rr Θ TT ll Θ TT. (5) HH ii zz ii, zz ii MM CC ii zz ii, zz ii zz ii + MM gg zz ii. (3) In order to use this control law, which is designed based on the feedback linearization control method, the function HH ii ( ) (i.e., CCCCC and ggggg) must be known. However, in practice these matrices may be unknown for most of real dynamical robots. To overcome this, we make use of an adaptive fuzzy logic system HH ii ( ) to approximate HH ii ( ).

6 6 ISRN Robotics erefore, using the singleton fuzzi er, product inference, and weighted average defuzzi er [38], the output of the fuzzy model can be expressed as where HH ii zz ii, zz ii θθ ii ζζtt zz ii, zz ii θθ + ζζ TT zz ii, zz ii θθ, (3) ζζ TT θθ ζζ TT ζζ θθ, θθ, ζζ TT nnnnnn θθ nnnnnn ζζ TT θθ ζζ TT ζζ θθ, θθ. ζζ TT nnnnnn θθ nnnnnn (3) Equation (3) suggests to us to rewrite the overall control law (9) as uu ii MM HH ii zz ii, zz ii θθ ii ff ii kk TT ee ii uu aaaa, (33) where uu aaaa is engaged to attenuate the fuzzy logic approximation error and external disturbances. A block diagram of the proposed control methodology is shown in Figure. Remark (ii ). In what follows, another main property of the proposed HH is introduced. It will be shown that the formation problem of multiagent system can be achieved even in the presence of dead-zone nonlinearities of the control actuators. Let us modify the dynamic model (9) as where MMzz ii + CC zz ii, zz ii zz ii + gg zz ii Φ uu ii + dd ii (tt) (34) φφ uu iii φφ uu iii Φ uu ii φφ uu (35) and φφφφφ φ φφ φ φφ represents the dead-zone function and can be expressed as mm (uu u uu) uu u uuu φφ (uu) bb b bb b bb, (36) mm (uu u uu) uu u uuuu where bb is the width of the dead-zone and mm is the slope of dead-zone line. e dead-zone parameters bb and mm are assumed to be bounded and the bounds of mm and bb are known as bb b [bb min, bb max ] and mm m mmm min, mm max ]. erefore (36) can be rewritten as φφ (uu) mmmm m mm (uu), mmmm mm m mmm (37) νν (uu) mmmm mmm m mm m mmm mmmm mm m mmmm From the aforementioned assumption on bounds of mm and bb, νννννν can be assumed bounded, (i.e., νννννν ν νν), where ρρ is the known upper bound that can be chosen as ρρ ρ ρρρρ max. By considering dd ii (ttt t tt ii (ttt t tttt ii ) and uu ii MMMM ii, where νν uu iii νν uu iii Υ uu ii, MM M MMMM nnnnn (38) νν nn uu then (34) can be rewritten as MMzz ii + CC zz ii, zz ii zz ii + gg zz ii uu ii + dd ii (tt), (39) which is the same as (9). erefore, we have proved that the proposed HH feedback controller is also robust to dead-zone input nonlinearities (36). 5. Stability Analysis is section presents the stability proof of the proposed novel adaptive fuzzy controller in (33). A Lyapunov candidate will be proposed and then an adaptation law and a robust compensator control input will be derived to satisfy the HH tracking performance in (5). To derive the adaptive law for adjusting θθ ii, we rst de ne the optimal parameter vector θθ ii as θθ ii arg min sup HH ii zzz zz z zz ii HH ii (zzz zz), θθ ii Ω and the minimum approximation error is de ned as (4) ww ii HH ii zz ii, zz ii HH ii zz ii, zz ii θθ ii, (4) where it can be assumed that ww ii LL [38]. By choosing the control input as [39] a er some manipulations, () can be rewritten as zz z ff ii HH ii zz ii, zz ii θθ ii HH ii zz ii, zz ii + kk ee ii + kk ee ii uu aaaa, (4) and the formation error dynamic can be expressed as ee ii HH ii zz ii, zz ii θθ ii HH ii zz ii, zz ii + kk ee ii + kk ee ii uu aaaa. (43) oreover by de ning EE ii [ee ii, ee ii, ee ii, ee ii,, ee nnnn, ee nnnn ] it is straightforward to write EE ii AAAA ii + BBBB aaaa + BB HH ii zz ii, zz ii HH ii zz ii, zz ii θθ ii, (44)

7 ISRN Robotics 7 u i M(z i ) z i + C(z i, z i ) z i + g(z i ) u i Dynamic of ith agent z i N N F L ij ( z i z j )+ N Q i ( z i ) i ji+ i Potential function f i z i u i M(Ĥ i (z i, z i θ i ) ḟi k T e i u ai ) Control input u ai r BT PE i Adaptation law E i e i (t) z i (t)+ t f i (τ)dτ Formation error evaluation e i, e i E i [e i, e i, e i, e i,..., e ni, e ni Formation error vector z i ζ T i E i θ i γζb T PE i Adaptation law ξ i Fuzzy rule base θ i ξ i ζi T (z i, z i )θ i M C i (z i, z i )z i M g(z i ) Fuzzy approximator Measured data FIGURE : Block diagram of the proposed adaptive fuzzy HH control scheme. where AA A AA nnnnn, BB B BB kk kk nnnnn TT. (45) Based on (3), (4), and (4), the matrix form of formation error in (44) can be rewritten as EE ii AAAA ii + BBBB aaaa + BBBB TT ii zz ii, zz ii θθ ii + BBBB ii, (46) where θθ ii θθ ii θθ ii. In the following theorem, it will be shown that the proposed control law (33) guarantees the stability and robustness of formation problem. eorem. Consider a group of fully autonomous agents with the dynamic represented in (8) and with the control law in (33). e robust compensator of iith robot uu aaaa and the fuzzy adaptation law are chosen as θθ γγγγ zz ii, zz ii BB TT PPPP ii, uu aaaa rr BBTT PPPP ii, (47) θθ γγγγ zz ii, zz ii BB TT PPPP ii, (48) where rr and γγ are positive constants and PP is the positive semide nite solution of following iccati li e e uation PPPP P PP TT PP P PP P rr PPPPPPTT PP P ρρ PPPPPPTT PP P PP (49) where is a positive semide nite matri and ρρ r. erefore, the HH tracking performance TT EE TT ii QQQQ ii dddd EE ii () TT PPPP ii () + γγ θθ() TT θθ () + γγ θθ() TT θθ () + ρρ TT ww TT ii ww ii dddd (5) can be achieved for a prescribed attenuation level ρρ and all the variables of closed loop system are bounded.

8 8 ISRN Robotics In order to derive the adaptive law for adjusting θθ ii, the Lyapunov candidate is chosen as VV V EETT ii PPPP ii + 4γγ θθ TT θθ + 4γγ θθ TT θθ. (5) + EE TT ii 4 PPPPPPTT zz ii, zz ii + θθ TT γγ θθ + ww TT ii BBTT PPPP ii + EE TT ii PPPPPP ii. (53) Using (46), the time derivative of VV is VV V EE TT ii PPPP ii + EETT ii PP EE ii + γγ θθ, + θθ TT θθ γγ + θθ γγ TT θθ θθ TT θθ + θθ γγ TT EE TT ii AATT PPPP ii + uu TT aaaa BBTT PPPP ii + θθ TT ζζ zz ii, zz ii BB TT PPPP ii + θθ TT ζζ zz ii, zz ii BB TT PPPP ii + ww TT ii BBTT PPPP ii + EE TT ii PPPPPP ii + EE TT ii PPPPPP aaaa + EETT ii PPPPPPTT zz ii, zz ii θθ +EE TT ii PPPPPPTT zz ii, zz ii θθ + EE TT ii PPPPPP ii + θθ TT θθ 4 γγ + θθ γγ TT θθ + θθ TT θθ γγ + θθ γγ TT θθ. (5) Substituting (47) in (5) and using the fact that θθ θθ, we get θθ θθ VV V EE TT ii AATT PPPP ii rr EETT ii PPPPPPTT PPPP ii + θθ TT ζζ zz ii, zz ii BB TT PPPP ii + θθ TT ζζ zz ii, zz ii BB TT PPPP ii + ww TT ii BBTT PPPP ii + EE TT ii PPPPPP ii rr EETT ii PPPPPPTT PPPP ii + EETT ii PPPPPPTT zz ii, zz ii θθ + EETT ii PPPPPPTT zz ii, zz ii θθ + EE TT ii PPPPPP ii + θθ TT 4 γγ θθ + θθ γγ TT θθ + θθ TT γγ θθ + θθ γγ TT θθ EE TT ii AA TT PP P PPPP P rr PPBBBBTT PP EE ii + EE TT ii 4 PPPPPPTT zz ii, zz ii + θθ TT γγ θθ Using adaptation law (48) and the Riccati-like equation (49), the above equation becomes VV V EE TT ii QQQQ ii ρρ EETT ii PPPPPPTT PPPP ii + ww TT ii BBTT PPPP ii + EE TT ii PPPPPP ii EE TT ii QQQQ i TT ρρ BBTT PPPP ii ρρρρ ii ρρ BBTT PPPP ii ρρρρ ii + ρρ ww TT ii ww ii EE TT ii QQQQ ii + ρρ ww TT ii ww ii. (54) Integrating the above inequality from tt t t to TT yields to VV (TT) VV () TT EE TT TT ii QQQQ ii dddd d dd ww TT ii ww ii dddd. (55) Using the fact that VVVVVV V V and from (49), the inequality TT EE TT ii QQQQ ii dddd EE ii () TT PPPP ii () + θθ() γγ TT θθ () + θθ() γγ TT θθ () + ρρ TT ww TT ii ww ii dddd (56) is obtained. erefore, the HH tracking equation (5) can be achieved and the proof is completed. 6. Simulation Results is section presents four simulation examples to illustrate the effectiveness of the proposed control scheme. In the rst example, a group of six agents with known dynamics as in (8) is considered. e second example presents the hexagonal formation of six partially unknown agents and an adaptive fuzzy logic system is used to approximate the unknown dynamics. is example proves the system stability under the proposed novel controller. In order to prove the

9 ISRN Robotics 9 TABLE : arameter speci cations of hexagonal formation. ii i iii i i iii i iii i i iii i iii i i iii i iii i i iii i iii i i dd TABLE : Agents initial positions. Agent number xx yy controller robustness, in the third example a white Gaussian noise is applied to all measured data and one of the agents is forced to be stationary and still the formation maintains its stabilizing performance. In the forth example one of the agents is chosen as the leader with a constant velocity. It is shown that proposed controller is able to form a dynamic D moving hexagon which tracks the leader. All the simulation results are implemented in MATLAB with. secs as the stepsize. e uni ue formation problem used in all ve simulation examples is a D hexagon with unit radius de ned by 5 6 jjjjjjj FF F zz ii zz jj dd, (57) where dd is speci ed in Table. In addition six random points in the D space are chosen to be the initial positions for six agents. ese points are assumed to be xed in all ve numerical simulations (Table ). 6.. Example I (Six Agents with Known Dynamics). Consider a group of six mobile agents with known dynamic models. Based on general model represented in (8), the nonlinear dynamic of the iith robot is considered as.6 xx ii+.5 xx ii+..6 yyii.5 yyii. uu ii, and a er some manipulations we get sgn xx ii sgn yy ii (58) xx ii.5 xx ii.33 sgn xx ii yyii.5 yyii.33 sgn yy ii uu ii. (59) To give a solution for the formation problem, formation error is de ned as (6) and the control law is designed based on (9), where kk 5 and kk 4. Figure 3(a) shows the formation trajectory of six robots starting from initial y (m) F (t) x (m) (a) Formation potential function Time (s) (b) FIGURE 3: Hexagonal formation of six agents with known dynamics. (a) Formation trajectory. (b) Formation potential. conditions (Table ) to the nal unit hexagon (57) in 3 secs and Figure 3(b) shows the potential value. e rst sub gure (Figure 3(a)) shows how smooth the controller guides all the agents to form the desired hexagon. is geometric formation does not have any xed position or direction, and it will only be determined by the agents initial position. e second sub- gure (Figure 3(b)) illustrates the potential decrement through the time. It is shown that the potential is forced to get stabilized in less than secs and it will be shown that the settling time for next simulation examples will more than secs. 6.. Example II (Six Agents with Partially Unknown Dynamics). To verify the effectiveness of proposed method the same novel formation error and (6) are chosen, respectively. Consider a group of six agents with the same dynamic models as (59). However, to design the control law, the dynamic model of agents is assumed to be partially unknown (i.e., CCCCC and ggggg in (8) are unknown). erefore, six fuzzy logic approximators are designed to approximate the unknown dynamic, where each agent approximator just needs the current position and velocity of

10 ISRN Robotics y (m) y (m) F (t) x (m) (a) Formation potential function Time (s) (b) F (t) 3 x (m) (a) Formation potential function Time (s) (b) FIGURE 4: Hexagonal formation of six agents with partially unknown dynamics. (a) Formation trajectory. (b) Formation potential. FIGURE 5: Hexagonal formation in presence of db noise and one agent failure. (a) Formation trajectory. (b) Formation potential. itself. ree Gaussian membership functions with unit variance are de ned and all θθs are initialized from zero vectors. e learning rate in (48) is set to γγ γ γγ and the output of the fuzzy system is achieved by choosing singleton fuzzi cation, center average defuzzi cation, Mamdani implication in the rule base, and product inference engine [38]. Simulation results of the proposed adaptive fuzzy HH technique with agents initial positions as shown in Table are shown as following. e motion trajectory in the rst 3 secs is illustrated in Figure 4(a) and the formation potential (57) is shown to be stabilized in Figure 4(b) Example III (Formation Problem in Presence of Measurement Noise and Agent Failure). In this example the robustness of proposed controller in presence of measurement noise and agent failure will be proved. e proposed potential function (3) and gradient-based method proposed in Section are able to obtain the exact formation even in the case of one agent failure. erefore, in this example it will be shown that when Agent #3 (xx 3 (), yy 3 () +) is forced to be stationary with zero velocity, other agents move toward this agent to achieve the hexagon formation. In addition a white Gaussian noise with SNR db is applied to all the measured data. All of the model characteristics and controller designs are the same as previous example in Section 6.. Motion trajectory and formation potential (57) of the rst 9 secs of simulation are shown in Figures 5(a) and 5(b), respectively Example IV (Formation Problem While Tracking the Leader). Previous example illustrated the good performance of formation stabilization, while agent failure (i.e., one agent remains stationary). However, the structure of potential function explained in (3) suggests to exempt one agent from the control law designed in (33), and let it move freely as the leader [8]. erefore, to run a more general simulation than previous example where one agent was stationary, here one of the agents is chosen as the leader and moves with a constant speed to a prede ned direction. en it is anticipated that, a er some transient formation, the agents position achieves the hexagon form in (57). All the problem parameters and controller design are the same as previous examples (i.e., (57), (59) and (4)). Agent #4 (xx 4 () +, yy 4 () ) is chosen as the leader, with constant velocity as xx 4 +.3, yy 4.5. (6)

11 ISRN Robotics References F (t) y (m) (a) x (m) Formation potential function Time (s) (b) FIGURE 6: Moving hexagonal formation while tracking the leader. (a) Formation trajectory. (b) Formation potential. e motion trajectory and formation potential (57) are shown in Figures 6(a) and 6(b), respectively. Simulation results prove that by using the same control law as (33) even the moving formation can be achieved. It can be seen that the formation is achieved in about 7 secs; however, this numerical simulation contains negligible steady state error (Figure 6(b)). 7. Conclusion In this paper, the formation control problem of a class of multiagent systems with partially unknown dynamics was investigated. On the basis of the Lyapunov stability theory, a novel decentralized adaptive fuzzy controller with corresponding parameter update law was developed and the stability of the system was proved even in the case of external disturbances and input nonlinearities. All the theoretical results were veri ed by simulation examples and good performance of the proposed controller was shown even in the case of agent failure, presence of measurement noise, and even moving formations. [] A. Badawy and C. R. McInnes, Small spacecra formation using potential functions, Acta Astronautica, vol. 65, no. -, pp , 9. [] L. E. Barnes, M. A. Fields, and K. P. Valavanis, Swarm formation control utilizing elliptical surfaces and limiting functions, IEEE Transactions on Systems, Man, and Cybernetics, Part B, vol. 39, no. 6, pp , 9. [3] S. H. Kim, C. Park, and F. Harashima, A self-organized fuzzy controller for wheeled mobile robot using an evolutionary algorithm, IEEE Transactions on Industrial Electronics, vol. 48, no., pp ,. [4] Y. Chung, C. Park, and F. Harashima, A position control differential drive wheeled mobile robot, IEEE Transactions on Industrial Electronics, vol. 48, no. 4, pp ,. [5] J. M. Lee, K. Son, M. C. Lee, J. W. Choi, S. H. Han, and M. H. Lee, Localization of a mobile robot using the image of a moving object, IEEE Transactions on Industrial Electronics, vol. 5, no. 3, pp. 6 69, 3. [6] M. J. Er and C. Deng, Obstacle avoidance of a mobile robot using hybrid learning approach, IEEE Transactions on Industrial Electronics, vol. 5, no. 3, pp , 5. [7] D. Lee and W. Chung, Discrete-status-based localization for indoor service robots, IEEE Transactions on Industrial Electronics, vol. 53, no. 5, pp , 6. [8] J. H. Reif and H. Wang, Social potential elds: a distributed behavioral control for autonomous robots, Robotics and Autonomous Systems, vol. 7, no. 3, pp. 7 94, 999. [9] S. Kato, S. Tsugawa, K. Tokuda, T. Matsui, and H. Fujii, Vehicle control algorithms for cooperative driving with automated vehicles and intervehicle communications, IEEE Transactions on Intelligent Transportation Systems, vol. 3, no. 3, pp. 55 6,. [] B. A. White, A. Tsourdos, I. Ashokaraj, S. Subchan, and R. Zbikowski, Contaminant cloud boundary monitoring using network of UAV sensors, IEEE Sensors Journal, vol. 8, no., pp , 8. [] R. Sepulchre, D. A. Paley, and N. E. Leonard, Stabilization of planar collective motion: all-to-all communication, Institute of Electrical and Electronics Engineers. Transactions on Automatic Control, vol. 5, no. 5, pp. 8 84, 7. [] D. V. Dimarogonas and K. J. Kyriakopoulos, Connectedness preserving distributed swarm aggregation for multiple kinematic robots, IEEE Transactions on Robotics, vol. 4, no. 5, pp. 3 3, 8. [3] M. Proetzsch, T. Luksch, and K. Berns, Development of complex robotic systems using the behavior-based control architecture ibc, Robotics and Autonomous Systems, vol. 58, no., pp ,. [4] K. Peng and Y. Yang, Leader-following consensus problem with a varying-velocity leader and time-varying delays, Physica A, vol. 388, no. -3, pp. 93 8, 9. [5] H. Takahashi, H. Nishi, and K. Ohnishi, Autonomous decentralized control for formation of multiple mobile robots considering ability of robot, IEEE Transactions on Industrial Electronics, vol. 5, no. 6, pp. 7 79, 4. [6] M. Defoort, T. Floquet, A. Kökösy, and W. Perruquetti, Sliding-mode formation control for cooperative autonomous mobile robots, IEEE Transactions on Industrial Electronics, vol. 55, no., pp , 8.

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