adaptive fuzzy formation control for a nonholonomic robotic swarm

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Adaptive Fuzzy Formation Control

for a Nonholonomic Robotic Swarm

Faridoon Shabaninia

School of Electrical and Computer EngineeringShiraz University, Shiraz, Iran

Email: [email protected]

Seyed Hamid Reza Abbasi

School of Electrical and Computer EngineeringShiraz University, Shiraz, Iran

Email: hamid [email protected]

Abstract—In this paper, artificial potential functions are usedto design the formation control input for kinematic model of the robots and matrix manipulations are used to transform non-holonomic model of a differentially driven vehicle into equivalentholonomic one. The advantages of the proposed controller can belisted as robustness to input nonlinearity, external disturbances,model uncertainties and measurement noises. Simulation results

are demonstrated for a swarm formation problem of a group of six unicycles, illustrating the effective attenuation of approxima-tion error.

I. INTRODUCTION

Multi-agent systems are very interesting decentralized sys-

tems and have been studied extensively over the past years

[1]–[4]. These systems have the complex behavior usually

seen in large-scale systems, although each agent is associated

with simple dynamics. Therefore, the decentralized control of

multi-agent systems have received increased research attention

in recent years.

In this research, a partially unknown nonlinear dynamic

model is adopted to unicycles and an adaptive fuzzy approxi-mator [5] is combined with H ∞ control technique [6], [7] to

propose a novel decentralized adaptive fuzzy formation control

methodology, with robust characteristics. The main advantage

of this control strategy is insensitivity to robot dynamic

uncertainties, external disturbances and input nonlinearities,

where control laws are applicable to nonholonomic robots (e.g.

a unicycle).

I I . PROBLEM FORMULATION

Consider a group of 2D point massless agents, where the

kinematic of the ith agent is considered as

zi = ui i ∈ {1, 2,...,N } , (1)

in which, zi ∈ R2 is the coordinate matrix (for a robot with

2-DOF) and ui ∈ R2 denotes the control inputs.

Then, consider the pair-wise potential fields, which are

defined between agents as

F ij = Lij (|zi − zj |) , ∀i, j ∈ {1, 2,...,N } , (2)

where Lij is designed to define a proper inter-agent potential

function.

By using the gradient descent method one can rewrite (1

as

zi = −f i = −∂F

∂zi, ∀i ∈ {1, 2,...,N } (3

which can be rewritten in the matrix form as Z = −∇F

where Z = [z1, z2,...,zN ] is the overall generalized coordinate

vector.At this stage, consider a more general form for each agen

as

ri =

vi cos θivi sin θi

θi = wi (4

where ri = (xi, yi)T

is the coordination of ith robot, vi and w

represent the linear and angular velocities. In order to include

the dynamic model of unicycle, equations

vi =1

mi

F i,

wi =

1

J i τ i (5

should be added to Eq. (4), where mi is the ith vehicle

mass, J i is the ith vehicle moment of intertia, and F i and τ

are the force and angular torque applied to the ith unicycle

respectively.

Therefore we can rewrite (4) and (5) in the form

gi (vi, ωi, θi) = −Γ−1i

−viωi sin(θi) − Liω

2

i cos(θi)viωi cos(θi) − Liωi

2 sin(θi)

(

and

ui = (F i τ i)T . (7

To find a direct relation between holonomic point an

mobile robot center, mobile robot position (ri), linear velocity(vi) and angular velocity (ωi) can be described as⎛

⎜⎜⎝rxiryiviwi

⎞⎟⎟⎠ =

⎛⎜⎜⎝

qxi − Li cos(θi)qyi − Li sin(θi)qxi cos(θi) + qyi sin(θi)− 1

Li

qxi cos(θi) + 1

Li

qyi cos(θi)

⎞⎟⎟⎠ (8

Therefore, it will be straight forward to write the dynamic

model in its most general form as

M i(θi)qi + gi(qi, qi, θi) = ui (9

482IEEE IRI 2011, August 3-5, 2011, Las Vegas, Nevada, USA

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III. CONTROLLER DESIGN

Consider, the kinematic formation error for the ith robot as

ei(t) = zi(t) +

t0

f i(τ )dτ, (10)

where ei ∈ R2 and f i is the gradient of designed potential

function.

Our design goal is to propose an adaptive fuzzy controllerso that

ei + k1ei + k2ei = 0 (11)

is achieved, where k1 and k2 are chosen to make (11)

asymptotically stable.

To design the controller, consider the control law proposed

as

ui = G−1(θi|φGi)

H i(zi, zi, θi|φHi) − f i

−k1ei − k2ei + uai

(12)

where uai

is engaged to attenuate the fuzzy logic approxima-

tion error and external disturbances.

The robust compensator of ith robot uai and the fuzzy

adaptation laws are chosen as

uai = −1

rBT PE i, (13)

φH

= −γ 1ζ Hi (zi, zi, θi)BT PE i, (14)

φG

= +γ 2ζ Gi (θi)BT PE iu

T i , (15)

where r, γ 1 and γ 2 are positive constants and P is the positive

semidefinite solution of following Riccati-like equation:

PA + AT

P + Q −

2

rPBBT

P +

1

ρ2PBBT

P = 0 (16)

in which, Q is a positive semidefinite matrix and 2ρ2 ≥ r.

Therefore, the H ∞ tracking performance

N i=1

T 0E T i QE i dt

≤N

i=1

E i(0)T PE i(0) + 1

γθ(0)T i θi(0)

+N

i=1

ρ2 T 0wT i wi dt

(17)

can be achieved for a prescribed attenuation level ρ and all

the variables of closed loop system are bounded.

Proof: For the detailed proof we refer to our previous article

[8].

IV. SIMULATION RESULTS

Consider a group of six mobile agents with dynamic models.

The nonlinear dynamic of the ith robot is considered as the

model in (4)-(5), while mi = 0.2 and J i = 1.

To give a solution for the formation problem, formation

error is defined as (10) and the control law is designed based

on (12), where k1 = 15 and k2 = 4. Moreover, six fuzzy

logic approximators are designed to approximate the unknown

dynamic, where each agent approximator just needs the curren

position and velocity of itself.

The motion trajectory of robots by using the proposed

adaptive fuzzy H ∞ technique in the first 30secs are illustrated

in Figure 1.

−2 −1 0 1 2

−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

x(m)

y ( m )

Fig. 1. Hexagonal Formation of Six Agents with Unknown Dynamics.

V. CONCLUSION

In this paper, the formation control problem of a class o

multi-agent systems with partially unknown nonholonomic dy

namics was investigated. On the basis of the Lyapunov stability

theory, a novel decentralized adaptive fuzzy controller with

corresponding parameter update laws was developed and the

stability of the system was proved even in the case of externa

disturbances and input nonlinearities. All the theoretical result

were verified by simulation examples and good performance

of the proposed controller was shown even in the case of agenfailure and presence of measurement noises.

REFERENCES

[1] J. Reif and H. Wang, “Social potential fields: A distributed behavioracontrol for autonomous robots,” Robotics and Autonomous Systemsvol. 27, no. 3, pp. 171–194, 1999.

[2] V. Gazi, “Swarm aggregations using artificial potentials and sliding-modcontrol,” Robotics, IEEE Transactions on, vol. 21, no. 6, pp. 1208–12142005.

[3] K. Peng and Y. Yang, “Leader-following consensus problem with varying-velocity leader and time-varying delays,” Physica A: Statistica

Mechanics and its Applications, vol. 388, no. 2-3, pp. 193–208, 2009.[4] M. Proetzsch, T. Luksch, and K. Berns, “Development of complex roboti

systems using the behavior-based control architecture ib2c,” Robotics an Autonomous Systems, vol. 58, no. 1, pp. 46–67, 2010.

[5] L. Wang, A Course in Fuzzy Systems and Control. NJ: Prentice-HalEnglewood Cliffs, 1997.

[6] J. Doyle, K. Glover, P. Khargonekar, and B. Francis, “State-space solutions to standard h2 and h∞ control problems,” Automatic Control, IEEETransactions on, vol. 34, no. 1, pp. 831–874, 1989.

[7] B. S. Chen, T. Lee, , and J. Feng, “A nonlinear h∞ control design irobotic systems under parameter perturbation and external disturbance,

International Journal of Control, vol. 59, no. 2, pp. 439–461, 1994.[8] B. R. Sahraei and F. Shabaninia, “A robust h∞ control design for swarm

formation control of multi- agent systems: A decentralized adaptive fuzzyapproach,” in 3rd International Symposium on Resilient Control System(ISRCS10), Idaho Falls, ID, 2010, pp. 79 – 84.

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adaptive fuzzy formation control for a nonholonomic robotic swarm

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