RESPONSIBLE EIGENVALUE CONCEPT FOR CONSENSUS CONTROL OF THREEROBOTIC VEHICLES: HETEROGENEOUS COUPLING CASE

Wei Qiao Rifat Sipahi

Department of Mechanical and Industrial Engineering, Northeastern University, Boston, MA 02115.

ABSTRACTThree robotic vehicles are controlled on an x − y plane by

sending them commands from a host computer via a Blue-

tooth connection, while capturing their positions with a cam-

era/software system. The control loop delays are substantial,

and are artificially added to mimic a long-distance control

scenario. Our Responsible Eigenvalue concept and analytical

rightmost root analysis approach are used to design heteroge-

neous couplings for the robots, to both achieve consensus and

to reduce consensus reach time. Implementation of the cou-

plings in experiments demonstrate that there is a good match

with simulations.

1. PRELIMINARIES

1.1. I/O Feedback Linearization of Robot Model

We start studying the dynamics of the differential drive robot

(Figure 1), where x, y, and θ are respectively x coordinates, ycoordinates, and orientation of the midpoint between the two

wheels, Vl and Vr are respectively the velocity of the left and

right wheels, ω = dθ/dt is the angular velocity, and d is the

fixed distance between the two wheels. Due to the nonholo-

nomic nature of the robot, one could control only two output

variables at the same time, therefore one simplifying choice

made here is to control (x, y) position first, and then the ori-

entation, as was also proposed in [1].Here, we focus on only (x, y) coordination of the robots.

For this, we choose the system outputs as Z = [x+δ cos θ, y+δ sin θ]T with δ = d/2, and assuming that the robot has suf-ficiently small time constant, we start with a kinematic modelwith output vector Z = [Zx, Zy]

T ,

Z =1

2Γ(θ)U =

1

2

(cos θ + sin θ cos θ − sin θsin θ − cos θ cos θ + sin θ

)U. (1)

The control law that linearizes the above system can be de-

signed as

U = 2Γ(θ)−1R, (2)

R =[γx, γy

]T, (3)

This research has been supported in part by the award from the National

Science Foundation ECCS 0901442. Detailed results of this work is currently

under review in IEEE Control System Technology.

where R can be selected based on an appropriate consensus

algorithm, for instance, following from [2–4]. One major

challenge here is that the measurements may not be instan-

taneous, but delayed by τ seconds in U , and therefore the

design of R based on these delayed measurements should be

carefully performed.

( , )x y

Fig. 1. Coordinates of differential drive robot.

1.2. LTI system with Homogeneous Delay and Review ofResponsible Eigenvalue Concept

The vector R in (2)-(3) is chosen here based on the following

consensus dynamics with homogeneous delays [5–8], given

by,

xi(t) = ui(t), (4)

ui(t) =n∑

k=1,k �=i

aik[xk(t− τ)− xi(t− τ)

], (5)

where i = 1, . . . n, xi(t) is the state of the ith agent, constant

coupling strength aik is nonnegative, aik ≥ 0, and only when

aik > 0, we have that agent i receives the state information of

agent k. In (4)-(5), τ is the nonnegative constant input delay.

The reason we choose to utilize the above consensus dy-

namics is because we uncovered some interesting properties

regarding its stability and performance behavior with respect

to the delay parameter τ . We summarize these properties

first [2, 9–12]:

• Let the matrix A be defined as A = [aik]. The largest

delay τ∗ that the system can tolerate is determined by

either one of the real eigenvalue, or a pair of complex

271 ISCCSP 2014978-1-4799-2890-3/14/$31.00 ©2014 IEEE 321 ISCCSP 2014978-1-4799-2890-3/14/$31.00 ©2014 IEEE 2223 ISCCSP 2014978-1-4799-2890-3/14/$31.00 ©2014 IEEE 913 ISCCSP 2014978-1-4799-2890-3/14/$31.00 ©2014 IEEE 258 ISCCSP 2014978-1-4799-2890-3/14/$31.00 ©2014 IEEE

conjugate eigenvalue λk(A) of A. This eigenvalue is

called the Responsible Eigenvalue [2].

• If all λk(A), except the invariant eigenvalue λ1(A) = 0describing the consensus nature of the system, is stable,

and if |�(λk(A))| > |�(λk(A))|, ∀k = 2, . . . , n, then

it is guaranteed that rightmost eigenvalues of the system

will move away from the imaginary axis as τ : 0→ 0+.

That is, the rightmost roots will start at sk(τ = 0) = λk

and transition to sk(τ = 0+)→ λk−ε, 0 < �(ε)� 1.

• When the above conditions hold, it is guaranteed that

only sk roots that emanate from λk will always remain

as rightmost eigenvalue of the system for delays up to

the delay margin of the system, τ ∈ [0, τ∗).

2. CONSENSUS CONTROL OF THREE ROBOTSWITH HETEROGENEOUS COUPLINGS

In this section, the objective is to use the tools associated with

the properties listed in the previous section, to design the het-

erogeneous couplings of the three robots, to achieve consen-

sus in experiments.

2.1. MADnet Platform

The Multi Agent Delayed Network (MADnet) is an exper-

imental platform we designed to study consensus systems.

The MADnet system with the block diagram shown in Fig-

ure 2 comprises an overhead CCD camera, a host computer,

and three differential drive mobile robots that communicate

with the host computer through Bluetooth connection.

Overhead-Camera (delay)20 23 f /

Software Environment: Roborealm (delay)20-23 fps/s

1280p x 720p resolution

Image processing, programming

E-puck Robot

Communication w/ bluetoothCommunication w/ bluetooth

Mass=0.15kg, height=4.5cm,

2 step motors, max speed=0.12874 m/sBluetooth Broadcasting (Delay)

Fig. 2. The setup of the MADnet platform.

The system is subject to an inherent time delay τ inh be-

tween each pair of robots, which includes the camera latency,

image processing, decision making delay and bluetooth com-

munication delay. It is measured around 0.15 ± 0.04 sec de-

pending on the current CPU load and the number of robots

being controlled. However, in order to mimic a long-distance

transmission scenario, we purposely incorporated additional

delay amounts in the decision making process in the host

computer, by awaiting certain number of sampling periods

to elapse before releasing the commands to the three robots.

With this added delay, we found out that delay amounts in the

robots communications is negligibly different [13], and hence

we can assume homogeneous delays τ = τnet, allowing us to

utilize the consensus protocol introduced in (4)-(5).

2.2. Implementation of Consensus Control

The I/O feedback linearization discussed in Section 1.1 is

adapted with the consensus protocol in (4)-(5) where the

agent couplings are chosen to be heterogeneous based on RE

concept and rightmost root analysis.

C

AB

X

YZA=ZB=ZC (t )

ZA(t =0)

ZB(t =0)

ZC(t =0)

D

D

DD

dAB

dAC

Fig. 3. The three robots move on x-y plane in order to reach

and maintain a triangular formation. The meeting point is

unknown to the robots, and is autonomously configured based

on the consensus protocol commanding the robots to meet,

under delayed communication.

For the formation-rendezvous, three robots are com-manded to move on the x-y plane to form a triangular-shapeformation (shown in Figure 3). Notice that in order to formup a triangular-shape formation by using (1), the positioncoordinates of each robot discussed in Section 1.1 needs tobe shifted by certain amount to prevent collision, thus we re-select the Z coordinates of each robot as shown in Figure 3,where D = 150 pixels = 135.94 mm. That is, we shall seekto reach position consensus of the points ZA = ZB = ZC fort→∞. Yet this meeting point is unknown and is determinedautonomously by the robots. In this scenario, the kinematicmodel of robot A with the new outputs is written as,

ZA =

(cos θA + sin θA cos θA − sin θAsin θA − cos θA cos θA + sin θA

)(VAl

VAr

)

and the model of robot B and robot C are in the same form.

Remark 1 Linearization commonly used for control of vehi-cles [1] holds only when the measurement of θ(t) is instanta-neous, thus Γ(θ)−1 · Γ(θ) = I . In the case when only θ(t −τ) is available, then linearization cannot be established in astraightforward manner since, in general, Γ(θ(t− τ inh))−1 ·

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Γ(θ) = I . Due to hardware limitations in the MADnet plat-form, indeed τ inh = 0 in the experiments; it is measuredaround 0.15± 0.04 sec. We have shown in [13] that τ inh canbe treated as uncertainty, and is small enough (as θ does notvary too fast), and hence can be neglected.

In the sequel, we assume that the inherent delay τ inh in

measuring θ is zero. Hence, the system can be decoupled into

two linear consensus dynamics as,

Zx(t) = AZx(t− τnet), Zy(t) = BZy(t− τnet),

where Zx = [ZAx, ZBx, ZCx]T , Zy = [ZAy, ZBy, ZCy]

T .

For simplicity, we choose A = B in the following.

2.2.1. Heterogeneous Coupling Design Based on RE concept

Time delay in this example is set as τ = τ20,xsp, which means

we purposely add 20 sampling periods in the communica-

tion channel, corresponding to τ = 1.05 sec delay. We shall

use this delay amount in simulating the full nonlinear model

for comparison purposes. Following from [14], the heteroge-

neous agent couplings are designed as,

A = B =

⎛⎝−0.98 0.49 0.490.34 −0.71 0.370.11 0.78 −0.89

⎞⎠ , (6)

from which the Responsible Eigenvalue is identified as λ2,3 =−1.2893 ± 0.1624j, and the delay margin of the consensus

system is τ∗ = 1.1124.

For the experiments, we provide the following measure-

ments: the sum of the distances between the robots (shown as

error), given by√d2AB + d2AC (see Figure 4), trajectories on

x-y plane, velocity and angular velocity of the robots.

Fig. 4. Experimental results with τnet = τ20,xsp ≈ 1.05,

with the sum of the distances between robots (shown as error),

trajectories, velocity and angular velocity.

Simulations are carried out for comparison and to vali-

date the experimental results. Notice that in the simulations,

the full nonlinear model is utilized also considering the orien-

tation measurement delay, see Figure 5.

Fig. 5. Simulation results with τnet = 1.05, with the sum

of the distances between robots (shown as error), trajectories,

velocity and angular velocity, τ inh = 0.18.

2.2.2. Heterogeneous Coupling Design

We next keep the inter-agent delay the same τnet = 1.05 sec,

pick s2,3(τnet = 1.05) = −0.8 ± 0.25j as the special point

where s2,3 is the rightmost, and then using this point we back

calculate the eigenvalue as λk = −0.3615±0.0146, k = 2, 3,

see details in [14, 15]. Since λ1 = 0 as per the consensus

requirement, its eigenvector is (1, 1, 1)T . Hence, using this

eigenvalue and its eigenvector in Gram-Schmidt process, we

can construct two orthogonal eigenvectors corresponding to

λ2,3, by which we can recover the designed system with new

coupling strengths given by:

A = B =

⎛⎝−0.25 0.1 0.150.13 −0.225 0.090.12 0.125 −0.245

⎞⎠ . (7)

In this case, the settling time of the consensus with delay

τnet = 1.05 can be estimated theoretically as 4/|�(s2,3)| = 5sec, with |�(s2,3)| = 0.8 being the distance of rightmost root

of the system to the imaginary axis (excluding λ1 = 0, which

determines only the “consensus” nature of the dynamics).

The simulation results based on this design are shown in

Figure 6 for comparison. The settling time in the simulation

is much larger, around 10 sec, consistent with the experimen-

tal results shown in Figure 7. The estimated settling time

4/|�(sk)| = 5 is in the same order of magnitude, but dif-

fers as it is based on only the dominant poles of the system,

ignoring the remaining poles and the numerator dynamics in

the linearized system.

3. CONCLUSIONS

Our recent results on Responsible Eigenvalue and analytical

study of rightmost roots on a class of consensus system with

delays are implemented on an experimental system with het-

erogeneous agent couplings, showing satisfactory match with

simulations.

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Fig. 6. Simulation results of the designed system with the

geometric average of the distance between robots (shown as

error), trajectories, velocity and angular velocity.

Fig. 7. Experimental results of the designed system with the

geometric average of the distance between robots (shown as

error), trajectories, velocity and angular velocity.

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