An Adaptive HH ∞-Based Formation Control for Multirobot Systems

We describe a decentralized formation problem formultiple robots, where anHH 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 andHH 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 [1,2].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][4][5][6][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 [1,2,[8][9][10].
In general, a robotic formation problem is de�ned as the organization of a swarm of agents into a particular shape in a 2D 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 [11] and distributed [12].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 [13], leader-following [14][15][16], graph-theoretic [12], and virtual structure approaches [17,18].
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 [15][16][17][18].For example, Takahashi et al. [15] proposed an SMC-based formation control scheme for multiple mobile robots, using the leaderfollowing strategy, in which they de�ned some performance ISRN Robotics indexes, so that robots can be controlled according to their ability.Defoort et al. [16] 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. [17] 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 2D formation.
Recently,  ∞ 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 [19][20][21][22][23][24].In an  ∞ 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 ( ∞ attenuation constraint) [19].is goal is generally represented as a Linear Matrix Inequality (LMI) problem.
In the traditional  ∞ 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 [25].
Since Zadeh [26] initiated the fuzzy set theory, fuzzy logic systems (FLS) have been widely applied to many real world applications [27][28][29][30].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 (1) there is no acceptable mathematical model for the robot and (2) 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 (1) and (2) are true.Besides, FLS schemes have been widely used in motion control of single robots [31,32].Using FLS integrated with  ∞ control technique can improve the robustness of controller and ensures the stability [25].
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-2 fuzzy approximator is combined with  ∞ 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., [17]), 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 2 presents the system description, problem formulation, and potential function evaluation.An introduction of interval type-2 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., [1,2,8,18]).erefore, in Section 2.1 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 in which      is the coordinate matrix (for a robot with ndegrees of freedom) and      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 2.2 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 -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   =     −    , ∀,   , ,  , } , where where   de�nes the global potential of each agent.Finally, the following three assumptions for potential function are considered [17,18].
At the �rst step, to propose a solution for multiagent formation control, the steepest descent direction [8,17,18] is chosen as and the control law is proposed.By substituting (6) which can be rewritten in the matrix form as Ż   where     ,  2 , ...,   ] 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  ∞ 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., [17,18]).
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     z     , ż  ż         , where      is the coordinate matrix (for a robot with n-degrees of freedom);   )    is a symmetric positive de�nite matrix and represents the inertia coe�cients.  , ż )    is the matrix of centripetal, Coriolis, damping, and rolling resistance forces;   )    is an vector of gravitational forces and      denotes the control inputs.
In most practical control problems of multiagent systems the inertia matrix   ) is a known constant matrix independent of   .erefore, the following assumption is considered.Assumption 4.  is the inertia matrix of robots, which is assumed to be a known and constant matrix.
In the next sections, the dynamic of each single agent will be assumed to be in the form of (10).

Interval Type-2 Fuzzy Logic System
In this section, the interval type-2 fuzzy set and the inference of the type-2 fuzzy logic system will be presented.A type-2 fuzzy set in universal set  is denoted as   which is characterized by a type-2 membership function    ) in (12).e    ) can be referred to as a secondary membership function or referred to as a secondary set, which is a type-1 fuzzy set in 0, ].In (13),   ) is a secondary grade, which is the amplitude of a secondary membership function; that is, 0 ≤   ) ≤ .e domain of a secondary membership function is called the primary membership of .In (13),   is the primary membership of , where     ⊆ 0, ] for all   ;  is a fuzzy set in 0, ], rather than a crisp point in 0, ], When   )  , for all     ⊆ 0, ], then the secondary MFs are interval sets such that    ) in ( 13) can be called an interval type-2 MF. erefore, the type-2 fuzzy set can be rewritten as Also, a Gaussian primary MF with uncertain mean and �xed standard deviation having an interval type-2 secondary MF can be called an interval type-2 Gaussian MF (13).It can be expressed as It is obvious that the type-2 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    ( and    (, respectively.Hence, (13) can be reexpressed as A type-2 fuzzy logic system (FLS) is very similar to a type-1 FLS as shown in Figure 1; the major structure difference being that the defuzzi�er block of a type-1 FLS is replaced by the output processing block in a type-2 FLS which consists of type-reduction followed by defuzzi�cation.
ere are �ve main parts in a type-2 FLS: fuzzi�er, rule base, inference engine, type reducer, and defuzzi�er.A type-2 FLS is a mapping     →  1 .Aer defuzzi�cation, fuzzy inference, type reduction, and defuzzi�cation, a crisp output can be obtained.
Consider a type-2 FLS having  inputs  1   1 , … ,  1    and one output   .e type-2 fuzzy rule base consists of a collection of IF-THEN rules, as in the type-1 case.We assume there are  rules and the rule of a type-2 relation between the input space  1 ×  2 × ⋯ ×   and the output space  can be expressed as where     are antecedent type-2 sets (  1, 2, … , ) and    s are consequent type-2 sets.
e inference engine combines rules and gives a mapping from input type-2 fuzzy sets to output type-2 fuzzy sets.To achieve this process, we have to compute unions and intersection of type-2 sets, as well as compositions of type-2 relations.e output of inference engine block is a type-2 set.By using the extension principle of type-1 defuzzi�cation method, type-reduction takes us from type-2 output sets of the FLS to a type-1 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: where  cos is the interval set determined by two end points For illustrative purposes, we brie�y provide the computation procedure for   .Without loss of generality, assume the    s are arranged in ascending order, that is,  1  ≤  2  ≤ ⋯ ≤    . Step where (/2)         =   and  Θ   Θ    = Θ  .

Controller Design Methodology
In this section a novel formation error based on the integral of formation gradient (5) will be proposed.en, a robust  ∞ 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 th robot as where   ∈   ,   represents the coordinate vector of th robot in (10) and   is the gradient of potential function de�ned in (7).It is straightforward to write the �rst and second derivatives of (26) as ė () = ż () +   , ë () = z () + ḟ .
Our design goal is to propose an adaptive fuzzy controller so that ë +   ė +  2   = 0 (28) is achieved, where   and  2 are chosen to make (28) asymptotically stable.
To design the controller, consider the control law proposed as where     , ż  =       , ż  ż +       .
In order to use this control law, which is designed based on the feedback linearization control method, the function   (⋅) (i.e., (⋅) and (⋅)) 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    (⋅) to approximate   (⋅).erefore, using the singleton fuzzi�er, product inference, and weighted average defuzzi�er [38] Equation ( 31) suggests to us to rewrite the overall control law (29) as where   is engaged to attenuate the fuzzy logic approximation error and external disturbances.A block diagram of the proposed control methodology is shown in Figure 2.

Remark 1 (𝑖𝑖𝑛𝑛𝑖𝑖𝑢𝑢𝑖𝑖 𝑛𝑛𝑖𝑖𝑛𝑛𝑖𝑖𝑖𝑖𝑛𝑛𝑒𝑒𝑎𝑎𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖).
In what follows, another main property of the proposed  ∞ 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  z +    , ż  ż +     = Φ    +   () where where  is the width of the dead-zone and  is the slope of dead-zone line.( From the aforementioned assumption on bounds of  and , () can be assumed bounded, (i.e., ()  ), where  is the known upper bound that can be chosen as  =  max .By considering   () =   () + (  ) and   =   , where then ( 34) can be rewritten as which is the same as (9).erefore, we have proved that the proposed  ∞ feedback controller is also robust to dead-zone input nonlinearities (36).

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  ∞ tracking performance in (50).
To derive the adaptive law for adjusting   , we �rst de�ne the optimal parameter vector  *  as and the minimum approximation error is de�ned as where it can be assumed that     ∞ [38].
By choosing the control input as [39] aer some manipulations, (10)

Formation error evaluation
Formation error vector Potential function Adaptation law where Based on (31), (40), and (41), the matrix form of formation error in (44) can be rewritten as where       −  *  .In the following theorem, it will be shown that the proposed control law (33) guarantees the stability and robustness of formation problem.eorem 2. Consider a group of  fully autonomous agents with the dynamic represented in (8) and with the control law in (33).e robust compensator of th robot   and the fuzzy adaptation law are chosen as where  and  are positive constants and  is the positive semide�nite solution of following �iccati�li�e e�uation� where � is a positive semide�nite matri� and 2 2 ≥ r.
erefore, the  ∞ tracking performance can be achieved for a prescribed attenuation level  and all the variables of closed loop system are bounded.

ISRN Robotics
In order to derive the adaptive law for adjusting   , the Lyapunov candidate is chosen as  50) can be achieved and the proof is completed.

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 where   is speci�ed in Table 1.
In addition six random points in the 2D space are chosen to be the initial positions for six agents.ese points are assumed to be �xed in all �ve numerical simulations (Table 2).

Example I (Six Agents with Known Dynamics
To give a solution for the formation problem, formation error is de�ned as (26) and the control law is designed based on (29), where     and    . Figure 3(a) shows the formation trajectory of six robots starting from initial conditions (Table 2) to the �nal unit hexagon (57) in 30 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 20 secs and it will be shown that the settling time for next simulation examples will more than 20 secs.

Example II (Six Agents with Partially Unknown Dynamics).
To verify the effectiveness of proposed method the same novel formation error and (26) 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.,  and  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 itself.ree Gaussian membership functions with unit variance are de�ned and all s are initialized from zero vectors.e learning rate in (48) is set to    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  ∞ technique with agents initial positions as shown in Table 2 are shown as following.e motion trajectory in the �rst 30 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 2 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 ( 3 (0)  −,  3 (0)  +) 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  20 db is applied to all the measured data.All of the model characteristics and controller

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).Formation potential function 30 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 70 secs; however, this numerical simulation contains negligible steady state error (Figure 6(b)).

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.

F 2 :
Block diagram of the proposed adaptive fuzzy  ∞ control scheme.

T 1 :
�arameter speci�cations of hexagonal formation.| |   |  |   |  |   |  |   |  |    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 2D moving hexagon which tracks the leader.All the simulation results are implemented in MATLAB with 0.01 secs as the stepsize.euni�ue formation problem used in all �ve simulation examples is a D hexagon with unit radius de�ned by

F 3 :
Hexagonal formation of six agents with known dynamics.(a) Formation trajectory.(b) Formation potential.

F 4 :
Hexagonal formation of six agents with partially unknown dynamics.(a) Formation trajectory.(b) Formation potential.

F 5 :
Hexagonal formation in presence of 20 db noise and one agent failure.(a) Formation trajectory.(b) Formation potential.designs are the same as previous example in Section 6.2.Motion trajectory and formation potential (57) of the �rst 90 secs of simulation are shown in Figures 5(a) and 5(b), respectively.

F 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.
and   , and       [  ,       and    [   ,    ] are the centroid of the type-2 interval consequent set    .For any value    cos ,  can be expressed as   is the minimum associated only with    , and   is the maximum associated only with    .Note that   and   depend only on mixture of   or   values.erefore, the lemost point   and the right-most point   can be expressed as a fuzzy basis function (FBF) expansion, that is, 1. Compute   in (22) by initially setting     ( e procedure to compute   is similar to compute   .Just in Step 2, we determine  (      ), such that Step 5. Set  �  equal to  ��  and return to Step 2.e point to separate two sides by number R can be decided from the above algorithm, one side using lower �ring  ).In the meantime, we have   = [   , and Θ   =      .