Fig. I. Differential robot.

that represent the motion of each vehicle in the 2D plane as(see Fig I)

xi

yiθi

=

cos θi 0sin θi 00 1

�viwi

�+

�Δxi

Δyi

�, (3)

for i = 1,...,m, where (xi, yi) is the position of the i-th robotand θi is the heading angle of the robot in the inertial frame,vi and wi are the linear and angular velocities, respectively.Δix and Δiy represent unknown but bounded uncertaintyterms associated to Coriolis effects, Centripetal forces or badmeasurements. The position tracking error of the i-th robot isdefined as

ei =

�xdi

ydi

�−�xi

yi

�, (4)

where(xdi , y

di ) is the desired planar position coordinate. Simi-

lar to [2], the synchronization error of the i-th robot is definedas:

εi = ciei − ci+1ei+1, (5)

where εi = [εxi εyi]T and coefficients ci which may be time

variant. Besides, when i = m, i + 1 is set to 1. Besides, ifthe synchronization error εi = 0, for all i=1,2,...,m , then thesynchronization goal is achieved. In order to link the positionand synchronization errors into one, the coupling error of thei-th robot is defined as [2]

Ei = ciei + β

� t

0

(εi − εi−1)dt, (6)

where Ei = [Exi Eyi]T and β is a positive diagonal matrix

with components βx and βy . In fact, the coupling error feedsback the information of the two neighboring robots i − 1and i+ 1, and gives specific weights to the formation versusconvergence. Also, when i = 1, i− 1 is set to m.In accordance to the sliding mode technique [11], a switchingfunction σi is proposed that allows to have asymptotic con-vergence of the position and synchronization errors to zero,

despite the presence of the uncertainty terms Δxi and Δyi.Such a switching function is chosen as

σi = Ei +Δ

� t

0

Eidt, (7)

where σi = [σxi σyi]T and Δ is a positive diagonal matrix

with components Δx and Δy .Thus in ideal sliding mode (when the dynamics of the mrobots is restricted to σi = 0 , i=1,...,m), with the componentsof the matrix Δ adequately chosen,Ei converges to zero,this is position and syncrhonization erros converge to zeroasymptotically.In order to attract the dynamics of the m robots to the i-thsurfaces σi = 0, i=1,...,m , the fractional order time derivativeof the function σi is set as

σ(1+α)i = −ksign(σi), (8)

where α is a real number, in accordance to the definition of afractional order derivative given in section II, k is a positivediagonal matrix with components kx and ky , and sign(σi) =[sign(σxi) sign(σyi)]

T with sign(·) being the sign function

sign(σi) =

�1 if σi > 0−1 if σi < 0

�. (9)

Differentiating equation (8) with respect to (−α), which isequivalent to integrate (8) to the order α [15], results in theexpression

σi = −k�sign(σi)

(−α)�. (10)

On the other hand, from equation (7) one has

σi = Ei +ΔEi, (11)

Fig. II. Forward point γi in a (2,0) differential robot.

whereEi = ciei + ciei + β(εi − εi−1). (12)

Substituing (12) into (11). Together with the i-th robot equa-tion model (3), and neglecting the uncertainty terms (this is,setting Δxi = Δyi = 0), leads to

σi = ciei + ci

�xdi

ydi

�− ciMi(θi)

�viwi

�+

+ β(εi − εi−1) +ΔEi = −k�sign(σi)

−α�, (13)

where the matrix

Mi(θi) =

�cos θi 0sin θi 0

�, (14)

is singular. Thus, a unique control [vi wi]T can not be ob-

tained,so that equation (8) or, equivalently, equation (10),holds. To circunvent such a singularity, the kinematics ofthe forward point γi = (pi, qi) for each mobile robot isconsidered, as it is shown in Fig.II. The coordinates of theforward point for each robot are given by

γi =

�piqi

�=

�xi + l cos θiqi + l sin θi

�, (15)

where l is the distance from the forward point to the midpointof the axle between the robot wheels. The dynamics of theforward point γi is then given by

Mi =

�piqi

�= Mγi

(θi)

�viwi

�+

�Δxi

Δyi

�, (16)

where the matrix,

Mγi(θi) =

�cos θi −l sin θisin θi l cos θi

�(17)

is nonsingular for any value of θi. Defining now the positiontracking error eγi

and the synchronization error εγias

eγi =

�xdi

ydi

�−�piqi

�, (18)

εγi= cieγi

− ci+1eγ,i+1, (19)

where eγi = [eγxi eγyi ]T , together with the coupling error

Eγi= cieγi

+ β

� t

o

(εγi− εγ,i−1) dτ, (20)

where Eγi= [Eγxi

Eγyi]T with coefficients ci and the matrix

β defined as before, it is possible to design a fractionalorder sliding mode controller that allows to have positionand synchronization convergence to zero by means of theswitching function

σγi = Eγi +Δ

� t

0

Eγi(τ)dτ, (21)

where σγi= [σγxi

σγyi]T and the matrix Δ is defined as

before.Setting the (1 + α) order derivative of σi as

σ1+αγi

= −k sign(σγi), (22)

with α a real number, sign(σγi) = [sign(σγxi) sign(σγyi)]T

and matrix k defined as previously, one gets the followingfractional order sliding mode controller

�viwi

�=

1

ciM−1

γi(θi)

�cieγi

+ ci

�xdi

ydi

�+ β(εi − εi−1)

�+

+1

ciM−1

γi(θi)

�ΔEγi

+ k�sign(σγi

)−α��

. (23)

This controller allows to attract the motion of i-th robotsto the surfaces σγi = 0 in finite time, where position andsynchronization convergence to zero occurs.Assuming that the bounds on the uncertainty terms are known,it is possible to give a sufficient condition that assues theattractiveness to the surface σγi

= 0 despite the presenceof such disturbances. Such an analysis is presented in thefollowing section.

IV. EXISTENCE CONDITIONS

In order to study the existence of the sliding mode whenthe fractional order sliding mode controller (23) is fedbackinto the dynamics (16)-(17), the following assumption on theboudedness of the uncertainty terms is made.Assumption 1.The uncertain terms are unknown smooth func-tions that satisfy

|Δxi| ≤ ηi1, |Δyi| ≤ ηi2, (24)

with: ηi1, ηi2 being nonzero positive real numbers which areknown. �

The following result gives a sufficient condition which assuresthe attraction of the perturbed dynamics to the surfaces σi = 0under assumption 1.

Theorem 1. Consider the perturbed dynamics (16)-(17) to-gether with assumption 1. If the elements of the matrix k, kxand ky , the coefficients ci and the bound ni = ni1+ni2 satisfy

kx

���sign(σγxi)(−α)

���+ ky

���sign(σγyi)(−α)

��� > cini. (25)

then, the fractional order sliding mode controller (23) assuresthe attraction of the perturbed dynamics to sliding surfacesσγi

= 0 �

Proof The following Lyapunov function candidate is proposed:

Vi(σγi) =1

2σTγiσγi

, (26)

which is positive definite. The time derivative of Vi(σγi) isgiven by

Vi = σTγiσγi = σT

γi[cieγi + β(εγi − εγ,i−1) +ΔEγi ] . (27)

Substituting the perturbed dynamics (16)-(17) into (27) to-gether with the fractional order sliding mode controller (23)gives

Vi = σTγi

�−ksign(σγi

)(−α) − ciΔp

�(28)

where Δp = [Δxi Δyi]T . On the other hand,

one can write σMxi and σMyi as σγxi =|σγxi | sign(σγxi),σγyi =

��σγyi

�� sign(σγyi), and since,0 < α < 1, sign

�sign(σγxi)

(−α)�

= sign(σγxi) andsign

�sign(σγyi)

−α�

= sign(σγyi) [16]. Thus, V can berewritten as

V = −k |σγi |T���sign(σγi)

(−α)���− ciσ

TγiΔp, (29)

where |σγi|T =

�|σγxi

|��σγyi

��� and��sign(σγi

)(−α)�� =���sign(σγxi

)(−α)�� ��sign(σγyi

)−α���. Finally, when majoring

(29) together with assumption 1, one has that

Vi ≤ − |σγi|T

�k���sign(σγi

)(−α)���− cini

�. (30)

Thus if the elements of the matrix k are such that (25) holds,Vi(σi) < 0 and the convergence to the surfaces σi = 0 isassured. �Remark 1. The fractional order sliding mode controller(FOSMC) (22) compensates the effect of the uncertainty termsΔxi and Δyi in the dynamics (16). The fact of defining theswitching function (21) (and the sliding surfaces σi = 0 ),together with its (1+α) fractional order derivative, makes thediscontinuos control active when the uncertainty terms appear(when σi �= 0), rejecting them.

V. EXPERIMENTAL RESULTS

The platform used to evaluate the performance of the frac-tional order sliding mode controller (23) consists of a setof three third generation Turtle® robots from ROBOTIS®.The programming of these robots is based on ROS (RobotOperating System), for Linux. A master program is used tolink the control signals with the robots by means of a pro-gram developed in Python (http://emanual.robotis.com/docs/e-n/platform/turtlebot3/overview/). Also, an OptiTrack® abso-lute localization system from Natural Point®, with four Flex-3® cameras and an OptiHub-2® were used. Before imple-menting the controller in this platform, this was programmedand evaluated in the simulation software GAZEBO®.The trajectory to be tracked by the 3 robots was a circle andthe parameters used, together with the corresponding initialconditions are given in table I and II. A fractional orderα = 0.5 was chosen and the results obtained were comparedwith those of an integer order α = 1.0. Fig. III shows thetrajectory tracking of the three robots in the plane ( withd1,d2,d3 being the desire trajectory for each one) while figuresIV to XI show the linear and angular velocities of each robotas well as the corresponding tracking errors, coupling errorsand synchronization errors. From the results shown one canobserve that a wobble appears during the tracking of thedesired path for both, the integer and the fractional ordercontroller. The tracking error is smaller when the fractionalorder controller is used though the third robot presents thegreatest difficulties when following the trajectory as well assynchronization ( this is also the case for the integer ordercontroller). An important feature of the fractional order slidingmode controller is that chattering is reduced in the controlsignals when compared to the integer order sliding modecontroller.

TABLE IPARAMETERS

Parameters Mobile robot1 2 3

Δx 8.0 8.0 8.0Δy 8.0 8.0 8.0βx 0.8 0.8 0.8βy 0.8 0.8 0.8kx 2.0 2.0 2.0ky 2.0 2.0 2.0cx 8.0 8.0 8.0cy 8.0 8.0 8.0la 0.08

ameter m. Δx and Δy are only used in GAZEBO.

TABLE IIINITIAL CONDITIONS OF THE PATH

Initial conditions Mobile robotof the path. 1 2 3

Position in xa 0.4 0.7 1.0Position in ya 0.0 0.0 0.0Position in θb 2.0 2.0 2.0

ameter m.bradian rad.

VI. CONCLUSIONS

A fractional order sliding mode controller was proposed in thiswork for the trajectory tracking fo a set of (2,0) differentialmobile robots while keeping a formation with syncrhonization.The controller was experimentally evaluated showing a goodperformance. In particular, the chattering in the control signals(linear and angular velocities) was reduced. Due to the largeamount of control gains needed in the controller proposed,further research is being carried out to find a procedure toadjust these gains in an optimal way as well as the bestfractional derivation order.

Fig. III. Trajectory tracking of the robots in the plane.

Fig. IV. Linear velocities of the robots.

Fig. V. angular velocities of the robots.

ACKNOWLEDGMENT

The first author aknowledges the support given by ConsejoNacional de Ciencia y Tecnologıa CONACYT,Mexico andCentro de Investigacion y Estudios Avanzados del I.P.N.CINVESTAV, Mexico

REFERENCES

[1] Dong Sun. Synchronization and Control of Multiagent Systems. CRCPress, 2011

[2] Dong Sun, Can Wang, Wen Shang, and Gang Feng. A synchronizationapproach to trajectory tracking of multiple mobile robots while maintain-ing time-varying formations. IEEE Transactions on Robotics, 25:10741086, 2009.

[3] Mujunder A., kurode S. B. Fractional-order sliding mode control forsingle link flexible manipulator. IEEE International Conference onControl Applications,2013, DOI: 10.1109/CCA.2013.6662773.

[4] Tang Y.G., Wang Y., Han M.Y. Adaptative fuzzy fractional-order slidingmode controller design for antilock braking systems. Neurocomput-ing,2016,DOI:10.1115/1.4032555.

Fig. VI. Tracking error in x.

Fig. VII. Tracking error in y.

[5] Aghababa M.P. A fractional-order controller for vibra-tion suppression of uncertain structures. ISA Transac-tions,2013,DOI:https://doi.org/10.1016/j.isatra.2013.07.010.

[6] Oustaloup A., In From fractality to non integer derivation: A fundamen-tal idea for a new process control strategy.; Analysis and Optimizationof Systems, Paris, 2006, 53-64, DOI: 10.1007/BFb0042201.

[7] Lurie BJ., In Three parameter tunable tilt-integral derivative (TID)controller.; US Patent, US5371 670, 1994.

[8] Tavazoei M.S. and Tavakoli-Kakhki M., In Compensation by fractional-order phase-lead/lag compensators.;IET Control Theory Applications,2014, DOI: 10.1049/iet-cta.2013.0138.

[9] Agrawal C. and Chen Y.Q., In An approximate method for nu-merically solving fractional-order optimal control problems of gen-eral form;Computers and Mathematics with Applications, 2010, DOIhttps://doi.org/10.1016/j.camwa.2009.08.006.

[10] Aguila-Camacho Norelys, Duarte-Mermoud Manuel A., Fractional adap-tive control for automatic voltage regulator,ISA Transactions. 2013, DOIhttps://doi.org/10.1016/j.isatra.2013.06.005.

[11] Utkin, V. I., In Sliding modes in optimization and control prob-lems;Springer, New York, 1992, ISBN 978-3-642-84379-2.

[12] Fridman, L., An averaging approach to chattering. IEEE Transactions

Fig. VIII. Coupling error in x.

Fig. IX. Coupling error in y.

on Automatic Control 2001, 46, 1260-1265, DOI: 10.1109/9.940930[13] Dadras S., Momeni H.R. Fractional terminal sliding mode control

design for a class of dynamical systems with uncertainty. Commu-nications in Nonlinear Science and Numerical Simulation,2012, DOI:https://doi.org/10.1016/j.cnsns.2011.04.032.

[14] Tang Y.G. Wang Y. Han M.Y. Lian Q. Adaptive fuzzy fractional-order sliding mode controller design for antilock braking systems.Neurocomputing , 2016.

[15] Shantanu Das. Functional Fractional Calculus for System Identificationand Con-trols. Springer-Verlag Berlin Heidelberg, 2008.

[16] Onder-Efe Mehmet. Integral sliding mode control of a quadrotor withfractional order reaching dynamics. Transactions of the Institute ofMeasurement and Control,33(8):985-1003, 2011.

Fig. X. Synchronization error in x.

Fig. XI. Synchronization error in y.