A Guidance Law for a Mobile Robot forCoverage Applications: A Limited

Information Approach

Twinkle Tripathy ∗ Arpita Sinha ∗

∗ Interdisciplinary Programme in Systems and Control Engineering,IIT Bombay, Mumbai–400 076, India

e-mail: twinkle.tripathy@sc.iitb.ac.in, asinha@sc.iitb.ac.in

Abstract: In this paper a guidance law has been proposed to solve applications related tocoverage of a pre-defined area using a mobile robot. In many situations, it may not be possibleto measure all the information due to practical limitations or from computational point ofview. So, we have used only range information to solve the above specified problem. The targetpoint is considered to be stationary in this particular case. Now, for any initial condition forwhich the velocity vector of the robot is perpendicular to the line of sight with respect tothe stationary target, the autonomous agent displays interesting behaviours which have beenanalysed theoretically. All the analytical results obtained here have been validated throughMATLAB simulations also.

Keywords: Coverage problems, minimal information, autonomous mobile robots, unicyclemodel, root locus.

1. INTRODUCTION

Over the past few decades the problem of coverage ofan area by a mobile robot has been a very challengingproblem. It finds several applications these days. Mostcommon applications would be vacuum cleaning, painting,humanitarian demining, foraging, farm irrigation and soon. Some of the more crucial applications would be mon-itor the area around a stationary point like a building.The coverage problem refers to the problem of guiding anautonomous agent to cover an unknown or predefined area.Several techniques have been proposed to achieve coverage,which vary depending on the application.

Surveillance based systems, discussed in Dhillon et al.(2003), Lai et al. (2007) and Wang et al. (2008), usewireless sensor networks to solve the coverage problem.The sensor nodes are capable of behaving autonomously.In Dhillon et al. (2003) and Wang et al. (2008), thecoverage achieved is average, but more emphasis is laidon the placement of sensors for effective coverage; Dhillonet al. (2003) have based their work on fixed sensor nodes.Lai et al. (2007) discuss surveillance systems based onwireless sensor networks. In Lai et al. (2007), the deployedsensors are divided into disjoint subsets of sensors, orsensor covers, such that each sensor cover can cover alltargets and work in turns. A surveillance system based onwireless sensor networks has been discussed by Yan et al.(2003). The system provides a differentiated surveillancesystem which deploys sensors nodes in appropriate areasdepending on the degree of surveillance required. But, thewireless sensor networks suffer from the problem of energyrequirements which increase with the number of sensornodes used in the network.

There are several algorithms which are used to coverunknown areas. For example, a lot of work has beendone on map based coverage techniques (see Wong et al.(2003), Zelinsky et al. (1993), Stachniss et al. (2003) andRutishauser et al. (2009) and references therein) whichwork by dividing the area into cells and then assigningvalues to the cells based on the presence of obstacles orfree space. Wong et al. (2003) used topological maps tosolve the coverage problem and their results are verifiedby simulation tests which show that over 99% of thesurface is covered. The topological maps, represent theenvironment as graphs where landmarks are nodes, andedges represent the connectivity between the landmarks.The algorithm achieves coverage with single mobile agent.Zelinsky et al. (1993) also use maps for coverage but with adifferent approach. The solution ensures complete coverageby the use of a distance transform path planning method-ology. The solution was simulated and implemented onan autonomous robot called Yamabico to give satisfac-tory performance. Stachniss et al. (2003) introduced theconcept of coverage maps where each cell of a given gridcorresponds to the amount of a cell which is covered by anobstacle; the coverage maps are improvement of occupancygrids which are based on the assumption that each cell iseither occupied or free. The model presented in the paperallows updation of the coverage maps upon input obtainedfrom sensors. Rutishauser et al. Rutishauser et al. (2009)solve the problem of collaborative coverage using a swarmof networked miniature robots again by the use of gridbased methods. For a multi-agent system, the problem ofcoverage, addressed by Batavia et al. (2009), has beensolved with high accuracy for a semi- structured envi-ronment. The approach has the advantage of an operatordriving the outline of a desired coverage area as input toa coverage generation algorithm. Acar et al. (2001) have

Proceedings of the Third International Conference onAdvances in Control and Optimization of DynamicalSystems, Indian Institute of Technology Kanpur, India,March 13-15, 2014

Frcsh1.6

437

Rmax

Rmin

T

Fig. 1. Region to be covered

Fig. 2. Coverage obtained by an Autonomous Agent

solved the coverage problem by the decomposition of theenvironment, by the use of voronoi diagrams.

In this paper we have proposed a guidance strategy thatuses only range information. The transmission of mini-mal information reduces the bandwidth requirement. Thisleads to energy conservation. Priliminary results on theguidance strategy have been done by Tripathy et al.(2013). In the work by Tripathy et al. (2013), we haveshown that the coverage strategy is stable for every initialcondition. The problem discussed in this paper is formu-lated as follows. The autonomous agent is modeled as anunicycle. We assume, the region to be covered, is circularor annular around the target as shown in Fig. 1. Theagent uses only distance information to cover the annularregion. The annular region is specified by Rmin and Rmax. It is shown that Rmin and Rmax depend on the initialconditions. Using this, we have derived:

• The trajectory of the agent under the initial conditionfor which the velocity vector is perpendicular to theline of sight with respect to the target.• The initial conditions necessary to achieve coverage

of an entire circular disk of any radius.

This paper is organized as follows. In Section 2 and 3,we have analysed the problem for a particular situation.Simulations results are presented in Section 4 and Section5 concludes the paper and discusses the areas on whichfurther work can be done in future.

2. COVERAGE STRATEGY

The paper addresses the coverage problem as shown inFig. 1. Let the target be at (xT , yT ) and is stationary atany instant t > 0, position of agent is given by (x(t), y(t)),

Reference

v

A(x, y)

θα

T(xT , yT )

VT(xV T , yV T )

(1− ρ)r

ρr

Fig. 3. Engagement geometry

as shown in Fig. 3. The kinematics of agent is as shownbelow:

x(t) = v cosα(t) (1)

y(t) = v sinα(t) (2)

α(t) = u(t) (3)

where v is the linear velocity, which is a constant for ourcase, α(t) is the heading angle and u(t) is the controlinput. In Fig. 3, r(t) is the line-of-sight(LOS) distanceand θ(t) is the LOS angle. Since only range measurementis feasible, agent knows only instantaneous r(t) and notθ(t). Let V T denote a virtual point, called the virtualtarget which is at a distance ρr(t) from the target alongthe LOS, where ρ ∈ (0, 1). In the control law, the lateralacceleration of the autonomous agent is made proportionalto the distance between the virtual target and the agent,

given by(

(xV T − x(t))2

+ (yV T − y(t))2)1/2

= (1−ρ)r(t).

So, the algorithm is defined in terms of aM (t) as:

aM (t) = h(1− ρ)r(t) (4)

α(t) =aM (t)

v= kr(t) (5)

where h is the constant of proportionality between aM and

the distance, (1 − ρ)r, and k = h(1−ρ)v . The parameter h

is a system gain.

The concept of the virtual target increases the flexibilityof the algorithm as we get another parameter ρ to bevaried as needed. The algorithm generates various cov-erage patterns surrounding the stationary target. One ofthe patterns generated is shown in Fig. 2. In the nextsection, the control algorithm is analysed for area coverageapplications. Consider the Fig. 3, the LOS between theagent and the target can be charaterised as follows:

vr = r = −v cos(α− θ) (6)

vθ = rθ = −v sin(α− θ) (7)

Let us define φ = α− θ. By combining (5)-(7) and solvingthe integration, we establish a relationship between r andφ as,

vr sinφ+kr3

3= vr0 sinφ0 +

kr303

(8)

In the above equation, r0 and φ0 are the initial conditions.We define a variable K which contains all the termscorresponding to the initial conditions, such that,

438

vθ

vr

A(φ = 3π2)C(φ = π

2)

B(φ = π)

D(φ = 0)

Fig. 4. (vθ, vr) plot

vθ

vr

C

B

A

D

Fig. 5. Engagement tra-jectory 1 in (vθ, vr)space

vθ

vr

B

A

C

Fig. 6. Engagement tra-jectory 2 in (vθ, vr)space

K =

(3vr0 sin(φ0)

k+ r30

)(9)

r3 +3vr sinφ

k−K = 0 (10)

By setting (6) to zero, we observe that r can take itsmaximum and minimum values that is Rmin and Rmaxwhen φ equals either 3π

2 or π2 . Next we look into how the

kinematics varies in the (vθ, vr) plane. Upon squaring andadding (6) and (7), we find that the instantaneous vθ andvr lie on a circle in the (vθ, vr) plane as shown in Fig. 4.By the use of (6) and (7), we can calculate the value ofφ corresponding to each set of instantaneous vr and vθ.Point A corresponds to (vθ, vr) = (v, 0), thus, φ = 3π

2 .Similarly, we get that at B, φ = π, at C, φ = π

2 and atD, φ = 0. Hence, the clockwise movement about the circleshown in Fig. 4 gives the direction of increasing values φ.Combining (5) - (7), we get,

dφ

dt=k(K + 2r3)

3r2(11)

The rate of change of φ with respect to time varies depend-ing on the sign of K. Considering the two cases separately,we have:Case 1: K > 0The above equation indicates that for K > 0 we get φ > 0.This corresponds to clockwise movement in the (vθ, vr)plane as shown in Fig. 5. Now, vr > 0 in the path C−B−Awhich means instantaneous r increases in this path upto

point A and thereafter it decreases as vr < 0 in the pathA −D − C. Hence, Rmax occurs at A(φ = 3π

2 ) and Rminoccurs at C(φ = π

2 ).

Case 2: K 6 0vr and vθ are continuous functions of φ. By re-arranging

(10), we get sinφ = k(K−r3)3vr . Hence, from (7) for all values

of K 6 0, vθ > 0. This indicates that the engagementgeometry can exist only in the first and fourth quadrantsin the (vθ, vr) plane. So, the only feasible inflexion pointis φ = 3π

2 . Hence, considering the continuity of the vθ andvr functions, we conclude that the (vθ, vr) plot oscillatesabout φ = 3π

2 as shown in Fig. 6. Now, vr > 0 in the pathA− B − A which means instantaneous r increases in thispath till the second time it reaches point A. Thereafter, itdecreases as vr < 0 in the path A−C −A till it comes tothe point A for the second time. Hence, in this case, bothRmax and Rmin occur at A(φ = 3π

2 ).

Thus, by using the coverage control algorithm given in(5), we get two types of engagement trajectories in (vr, vθ)plane, depending on K or the initial conditions (r0, φ0). Bydifferentiating (6) and (7), we get:

vr = −vθφ (12)

vθ = vrφ (13)

At the points B and C shown in Fig. 6, vr and vθ be-come zero simultaneously, which implies φ = 0. Usingthis condition in (11), we get the following result for theinstantaneous value of r:

r =

(K ′

2

) 13

(14)

where K ′ = (−K). Substituting (14) in (10) and furthersolving, we get an expression of instantaneous φ as,

sinφ =−kv

(K ′ 2

4

) 13

(15)

Eqn. (15) gives a bound on K ′ = (−K) as sine functioncan take values between -1 to 1. Hence, the minimum valueof K can be obtained by setting sinφ = −1, which gives

K = −K ′ = −2√

v3

k3 . Let us denote the lower value of

K by: −Kbr = −2√

v3

k3 . Thus, coverage becomes feasible

for negative values of K only if −Kbr 6 K 6 0. Next weaddress two problems: If we are given an initial condition,what will be the coverage area and if we are given an areato be covered, what should be the initial conditions toachieve the coverage? Given an initial condition (r0, φ0),we have the following result:

Theorem 1. (Tripathy et al. (2013)) Given a set of initialconditions (r0, φ0) for an agent with kinematics (1)-(3) anda control law given by (5), coverage will be achieved for aregion given by the maximum radius Rmax and minimumradius Rmin which are given as:

Rmax = az1 + bz2 (16)

Rmin = az3 + bz4 (17)

439

Reference

v

A(x, y)

θα

φ = 3π2

T(xT , yT )

Fig. 7. Agent and target initial engagement for φ0 = 3π2

where

a =

−1 + i√

3

2, if −Kbr 6 K 6 0

1, if 0 < K <∞

b =

−1− i√

3

2, if −Kbr 6 K 6 0

1, if 0 < K <∞

z1 = −

(−K

2+

√K2

4− v3

k3

) 13

z2 = −

(−K

2−√K2

4− v3

k3

) 13

z3 =

z2 if −Kbr 6 K 6 0

−

(−K

2+

√K2

4+v3

k3

) 13

if 0 < K <∞

z4 =

z1 if −Kbr 6 K 6 0

−

(−K

2−√K2

4+v3

k3

) 13

if 0 < K <∞.

The detailed proof of the theorem is given in the work byTripathy et al. (2013).

3. AREA OF COVERAGE

In this section we have considered the special case whenthe initial conditions are selected such that φ0 = 3π

2 . Inphysical sense, this is the configuration when that theinitial heading angle is perpendicular to the LOS betweenthe agent and the stationary target as shown in fig. (7).

Critical Initial Position:

Critical initial position refers to that value of (r0, φ0) forwhich Rmax and Rmin are equal. This makes the agentmove on a circular path around the stationary target forall time. Also, it is interesting to note that for these initialcoordinates, we get Rmax = Rmin = r0.

Theorem 2. (Tripathy et al. (2013)) For r0 =√

vk and

φ0 = 3π2 , the trajectory of an agent with kinematics (1)-

(3) and a control law given by (5), is a circle of radius√vk .

The detailed proof of this theorem has been covered in thepaper by Tripathy et al. (2013).

Proposition 3. If φ0 = 3π2 and r0 ∈ [0,

√vk ], then Rmin =

r0.

Proof. Substituting φ0 = 3π2 in (9), we find out that for

r0 ∈ [0,√

3vk ] K 6 0 and otherwise K > 0. Let us consider

the case r0 ∈ [0,√

3vk ]. From Section 2, we know that

for K 6 0, Rmax and Rmin occur at φ = 3π2 . Now, we

substitute φ0 = 3π2 in (8) and solve it to get the following

roots:

r1 = r0 (18)

r2 =−r02

+

√−3r02

4+

3v

k(19)

One of the roots is Rmax and the other one is Rmin. Thenegative roots have been ignored.

r1 − r2 =3r02− 1

2

√−3r02 +

12v

k(20)

From (20) it can be shown that r1 − r2 6 0 for r0 6√

vk .

Hence, Rmin = r0 for r0 ∈ [0,√

vk ].

Proposition 4. If φ0 = 3π2 and r0 ∈ [ vk ,

3vk ], then Rmax =

r0.

Proof. In the proof given for Proposition 3, we have seenthat for r0 ∈ [0,

√vk ] K 6 0. Hence, in this region both

Rmax and Rmin occur at φ0 = 3π2 . So, substituting the

value of φ0 in (8), we get the roots (18) and (19). Again, wehave ignored the negative root. Using (20), it can be shown

that r1 − r2 > 0 for r0 ∈ [√

vk ,√

3vk ]. Hence, Rmax = r0

for r0 ∈ [√

vk ,√

3vk ].

Theorem 5. The agent covers a maximum radius of r0 ifit starts at r0 >

√vk with heading φ0 = 3π

2 .

Proof. For r0 ∈ [√

vk ,√

3vk ], it follows directly from

Proposition 4 that Rmax = r0. From Section 2, we knowthat Rmax always occurs at φ0 = 3π

2 irrespective of thesign of K. Substituting the value of φ0 in (8), we get thepositive roots as,

r1 = r0 (21)

r2 =−

(K

2+

√K2

4+v3

k3)

13 − (

K

2−√K2

4+v3

k3

) 13

(22)

Using simple calculations, it can be shown that r1 > r2 for

every r0 >√

3vk . Hence, we conclude that for r0 >

√vk ,

Rmax = r0.

440

Corollary 6. Given φ0 = 3π2 and r0 ∈ [0,

√3vk ], the

trajectory of the agent is bound by a circle of radius√

3vk .

Proof. In Proposition 4, we found out that for r0 ∈[√

vk ,√

3vk ] Rmax = r0. Hence, Rmax becomes maximum

when r0 is maximum. Hence, in the given range the

maximum value of Rmax is√

3vk . For the case when

r0 ∈ [0,√

vk ], the expression for Rmax is given by (19).

The first derivative of the expression is given by − 12 −

3r0(−3r02 + 12v

k )12 . The first derivative is always negative,

so Rmax decreases with respect to r0. Hence, Rmax ismaximum for r0 = 0. Using (19), the maximum value of

Rmax is equal to√

3vk . Since, for the given range of r0 the

maximum value of Rmax has an upper bound of√

3vk , the

trajectory is also bounded by a circle of radius√

3vk .

Theorem 7. To cover an entire region of radius√

3vk , the

agent must have an initial condition given by (d, 3π2 ) or

(0, 3π2 ) and k = 3vd2 .

Proof. In Corollary 6, we derived that the maximumvalue of Rmax is equal to 3v

k which occurs at r0 = 0 and√3vk . Using (18) and (19), we calculated that for r0 = 0

and√

3vk , the value of Rmin is equal to zero. This proves

that the agent covers the entire circular disk of radius 3vk

for the initial configurations ( 3vk ,

3π2 ) and (0, 3π2 ).

Let us consider a circle of radius d which is desired tobe covered. The value of the controller gain k should bechosen such that d =

√3vk. Hence, as explained in the

above paragraph, for k = 3vd2 the agent should start at

(d, 3π2 ) and (0, 3π2 ) in order to cover a circular area of radiusd.

4. SIMULATION RESULTS

In this section, we validate the results obtained in Section3 through MATLAB simulations. The simulations areperformed with the parameters h = 0.01, ρ = 0.3 andv = 5. The control algorithm developed in (5) leads toformation of patterns around the target ensuring coverageof the area surrounding the target.

4.1 Critical Initial Position

For initial conditions (√

vk ,

3π2 ), the trajectory of the agent

is circular with the target as the centre as given byTheorem 2. This result has been verified by simulationas shown in fig. (10)

4.2 Range of Rmax and Rmin

In Proposition 3, we derived that Rmin varies linearlywith respect to r0 for r0 ∈ [0,

√vk ]. This can be verified

by the fig. 11 which has been obtained using MATLAB

Fig. 8. Trajectory plot: r0 = 45, φ0 = 110π180

Fig. 9. r versus t : r0 = 45, φ0 = 110π180

Fig. 10. Agent trajectory for critical initial position

simulation. Similarly in Proposition 4, we found out that

Rmax varies linearly with respect to r0 for r0 ∈ [√

vk ,√

3vk ].

This result has been shown in fig. 12.

5. CONCLUSION

In the paper, we have analysed the control law (5) underinitial conditions such that the velocity vector is perpen-dicular to the line of sight with respect to the stationarytarget. For this initial heading, the agent exhibits sev-eral interesting behaviours. For the critical initial initialcondition given in Theorem 2, the agent circles aroundthe target. Hence, the agent can continuously monitor thetarget using only range information. The radius of the cir-cular trajectory depends on the controller gain and speedv and, hence, can be specified by the user. In Theorem 7,it has been shown that by adjusting the controller gain,

441

Fig. 11. Variation of Rmin with respect to r0

Fig. 12. Variation of Rmax with respect to r0

the agent can be made to cover an entire disk. The workcan be extended to find out the effect of the controllergain k and speed v. Also, more work can be done tomake the controller robust. Since, the agent uses onlyone(range) parameter’s data for the control law, the noiseassociated with the system reduces. The guidance law alsouses minimal information for coverage. This leads to costsaving. It also minimises the energy consumption which isone of the major issues of today!

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