Cooperative control of multiple vehicles with limited sensing

INTERNATIONAL JOURNAL OF ADAPTIVE CONTROL AND SIGNAL PROCESSINGInt. J. Adapt. Control Signal Process. 2007; 21:115–131Published online 18 September 2006 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/acs.921

Cooperative control of multiple vehicles with limited sensing

Jian Chen1,*,y, Darren M. Dawson2, Mohammad Salah2 and Timothy Burg2

1Department of Mechanical and Aeronautical Engineering, Clarkson University, Potsdam, NY 13699, U.S.A.2Department of Electrical and Computer Engineering, Clemson University, Clemson, SC 29634, U.S.A.

SUMMARY

A navigation function based cooperative control is developed in this paper for the navigation of multipleUnmanned Aerial Vehicles (UAVs) in the presence of known stationary obstacles and unknown enemyassets (EAs). Specifically, the motion of UAVs are planned in a centralized fashion. The standardnavigation function approach is extended to a multiple navigation strategy with an analytical switchamong different cases due to the limited sensing zone of the UAVs. A differentiable controller is proposedbased on this navigation function that yields asymptotic convergence. A discussion for avoiding movingEAs is presented. Simulation results are provided to illustrate the performance of the proposed controlstrategy. Copyright # 2006 John Wiley & Sons, Ltd.

Received 30 September 2005; Revised 9 June 2006; Accepted 3 July 2006

KEY WORDS: cooperative control; non-linear systems; multiple vehicles

1. INTRODUCTION

Numerous researchers have proposed algorithms to address the motion control problemassociated with robotic task execution in an obstacle cluttered environment. A comprehensivesummary of techniques that address the classic geometric problem of constructing a collision-free path and traditional path planning algorithms is provided in Section 9, ‘LiteratureLandmarks’, of Chapter 1 of Reference [1]. Since the pioneering work by Khatib Reference [2],it is clear that the construction and use of potential functions has continued to be one of themainstream approaches to robotic task execution in the presence of obstacles. In short, potentialfunctions produce a repulsive potential field around the robot workspace boundary andobstacles and an attractive potential field at the goal configuration. A comprehensive overviewof research directed at potential functions is provided in Reference [1]. One of the criticisms ofthe potential function approach is that local minima can occur that can cause the robot to ‘get

*Correspondence to: Jian Chen, 6757 State Highway 56, Apt. 9, Potsdam, New York 13676, U.S.A.yE-mail: [email protected]

Copyright # 2006 John Wiley & Sons, Ltd.

stuck’ without reaching the goal position. Several researchers have proposed approaches toaddress the local minima issue (e.g. see References [3–7]). One approach to address the localminima issue was provided by Koditschek in Reference [8] for holonomic systems (see alsoReferences [9, 10]) that is based on a special kind of potential function, coined a navigationfunction, that has a refined mathematical structure which guarantees a unique minimum exists.For other related work that has focused on the development of potential function-basedapproaches, the reader is referred to References [10–17].

By leveraging previous results directed at classic (single robot) systems, more recent researchhas focused on the development of potential function-based approaches for the morechallenging multiple agent system. For example, Loizou et al. [18] extended the navigationfunction methodology, established for single robot navigation, to the case of multiple robots.Loizou et al. [19] presented an extension to the navigation function methodology to the casewhere unmodelled obstacles are introduced in the workspace. The approach of [19] constructs apotential field that models all known environment features and combines it with an efficientcontrol scheme that handles additional unknown features. In Reference [20], gyroscopic forcesand scalar potentials were utilized to create swarming behaviours for multiple agent systems. Asstated in Reference [20], two main assumptions, the kinetic energy of the vehicle is bounded andonly one obstacle is present in the detection shell, limit the application of the approach. A decen-tralized motion control approach was proposed in Reference [21] in which the agent reacts withthe agents in its neighbourhood. No obstacles were considered in Reference [21]. In Reference[22], the flocking motion remains stable as long as the graph that describes the neighbouringrelations among the agents in the group is always connected. For local agent interaction, a non-smooth potential function was applied between two agents under a local interaction regime.Additionally, demonstration of the non-smooth control law in Reference [23] showed the use oflocal sensing (limited sensing distance) to affect motion but not affect the stability properties ofthe group. This type of system falls into the category of differential equations withdiscontinuous right-hand sides, and hence, the stability tools are generally based on non-smooth analysis. More recently, Olfati-Saber et al. presented a theoretical framework for thedesign and analysis of distributed flocking algorithms in the presence of multiple obstacles [24].

The goal of this paper is to develop a cooperative control algorithm for the multiple UAVproblem to ensure that all of the UAVs will: (i) remain in the combat zone, (ii) avoid collisionswith other UAVs, (iii) avoid collisions with predetermined stationary objects, (iv) avoid beingdetected by EAs, and (v) ensure each UAV moves to its desired location in the combat zone. Theunique aspect of this approach is that the UAVs only act with respect to the EAs within itssensing zone. That is, the navigation function is analytically switched when EAs enter thesensing zone of UAVs without corrupting the stability and convergence properties of thenavigation function. The proposed approach can be applied to numerous applicationsz such asmobile sensor network deployment, pre-flocking (gathering the UAVs to a specific formation ina certain location), and suppression of enemy air defense using a cooperative group of UAVs.

This paper is organized in the following manner. In Section 2, the problem is formulated forthe multiple UAVs navigation. In Section 3, a smooth bump function, a boundary function,obstacle functions, EA detection zone functions, and UAV collision functions are developed.In Section 4, a multiple UAV navigation function is designed, and a cooperative control

zAlthough the vehicle model in this manuscript is the UAV, it can be applied to any multiple agent system which hasvarious applications and has attracted significant attention as stated in Reference [22].

J. CHEN ET AL.116

Copyright # 2006 John Wiley & Sons, Ltd. Int. J. Adapt. Control Signal Process. 2007; 21:115–131

DOI: 10.1002/acs

algorithm is developed. A discussion for avoiding moving EAs is presented in Section 5.Simulation results illustrating the performance of the cooperative control algorithm are given inSection 6, and concluding remarks are given in Section 7.

2. PROBLEM FORMULATION

As illustrated by Figure 1, all UAVs are assumed to be initially in the combat zone (i.e. the largecircle), which is a configuration space with a boundary. In the combat zone, there are somestationary obstacles and EAs. All UAVs have an onboard sensor, which can only sense objectsin its neighbourhood denoted as the sensing zone (i.e. due to the physical limitation of theonboard sensing capability of each UAV). With a limited detection range, the EAs can also onlydetect objects in its neighbourhood denoted as the detection zone. The cooperative controlobjectives for the multiple UAVs are to ensure that all of the UAVs will:

* Objective 1: Remain in the combat zone.* Objective 2: Avoid collisions with other UAVs.* Objective 3: Avoid collisions with the predetermined stationary objects.* Objective 4: Avoid being detected by EAs.* Objective 5: Ensure that each UAV moves to its desired location in the combat zone.

To address the multiple UAV cooperative control problem in this paper, we use the followingassumptions:

* Assumption 1: The physical dimensions of the combat zone are assumed to be known by allof the UAVs.

Figure 1. Multiple UAVs in the combat zone.

COOPERATIVE CONTROL OF MULTIPLE VEHICLES 117


DOI: 10.1002/acs

* Assumption 2: The location and the physical dimensions of the stationary obstacles areknown within the combat zone by all of UAVs.

* Assumption 3: The UAV sensing zone and the EA detection zone are much greater than thesize of the EA; therefore, the EAs can be treated as points.

* Assumption 4: The detection zone of the EA is known by all of UAVs.* Assumption 5: The sensing zone of the UAVs is greater than the detection zone of the EAs

(otherwise it is impossible for the UAVs to avoid being detected by the EAs).* Assumption 6: Within the finite combat zone, all UAVs have access to all of the other

UAVs’ position.* Assumption 7: Initially, all of the UAVs are in a non-contact configuration in the combat zone

and do not contact the EAs, stationary obstacles, and the boundary of the combat zone.

3. MODEL DEVELOPMENT

To explain the approach, we first transform the physical geometry of each UAV into a spherewith qiðtÞ 2 R3 and rzi 2 R; i ¼ 1; 2; . . . ; n; denoting the time-varying centre and the constantradius of the ith UAV sphere, respectively, and n denotes the number of the UAVs. Thecomposite configuration for the UAV qðtÞ 2 R3n is defined as

qðtÞ ¼ ½qT1 ðtÞ qT2 ðtÞ � � � q

Tn ðtÞ�

T ð1Þ

We will assume that the multiple UAV system can be described by the following kinematicmodel:

’q ¼ u ð2Þ

where uðtÞ 2 R3n is the cooperative control input. The goal configuration of the UAVs is denotedas qn 2 R3n and defined by

qn ¼ ½q*T

1 q*T

2 � � � q*Tn �T ð3Þ

where qni 2 R3; i ¼ 1; 2; . . . ; n; denotes the desired constant location of the ith UAV.

3.1. Smooth bump function

The multiple UAV navigation problem is complicated by the fact that the sensing zone of theUAVs is finite. To deal with this problem, a smooth bump function is introduced to analyticallyswitch the repulsive force generated by the EAs and the boundary of the combat zone from zeroto one. Indeed, it is this type of analytical smoothing approach that allows the boundaryfunctions and EA detection functions developed subsequently to be appropriately smoothed outto facilitate the gradient-like differentiation that is utilized to construct the cooperative controlalgorithm.

Definition 1The smooth bump function, denoted by rhð�Þ : Rþ ! ½0; 1�; is a scalar function satisfying [25]

rhðxÞ ¼

1; 04x4h

0; x51

2 ð0; 1Þ; h5x51

8>><>>: ð4Þ

J. CHEN ET AL.118


DOI: 10.1002/acs

where Rþ denotes non-negative real number, and h 2 ð0; 1Þ is a positive constant parameter ofthe bump function.

The smooth property of the bump function defined in (4) is illustrated in Figure 2.Based on the definition of the bump function in (4), one possible choice is the following bump

function introduced in [25]

rhðxÞ ¼

1; x 2 ½0; h�

1

21þ cos p

x� h

1� h

� �� ; x 2 ðh; 1Þ

0 otherwise

8>>>><>>>>:

ð5Þ

3.2. Boundary function

To ensure that the UAVs remain within the predefined combat zone, a scalar boundary functionis introduced to determine the relationship between the boundary of the combat zone and theposition of the UAVs. The objective in selecting a boundary function is to prevent any UAVfrom contacting any part of the combat zone boundary.

Definition 2The boundary function, denoted by b0ð�Þ 2 R; is a function satisfying

b0 ¼

0At least one UAV contacts the boundary

of the combat zone

1

All UAV are inside the combat zone and

the boundary of the combat zone is

outside the sensing zone of UAVs

2 ð0; 1Þ Otherwise

8>>>>>>>>>>>><>>>>>>>>>>>>:

ð6Þ

Figure 2. Bump function.



DOI: 10.1002/acs

As indicated by (6), the function b0ð�Þ approaches zero if any of the UAVs approach theboundary of the combat zone, and hence, this type of design for b0ð�Þ spawns a gradient-liketerm to ensure that the UAV moves away from the boundary of the combat zone. Based on (4)and (6), the boundary function can be designed as follows:

b0 ¼Yni¼1

rh0i1

ro0 � rzijjqi � qo0jj

� �ð7Þ

where qo0 2 R3 denotes the position of the centre of the combat zone, and ro0 2 R

denotes the radius of the combat zone. From (6) and (7), it is clear that: (i) b0 ¼ 0when for any i; jjqi � qo0jj5ro0 � rzi; and (ii) b0 ¼ 1 only when for all i; jjqi � qo0jj4h0iðro0 � rziÞ: Based on (4), (6), and (7), the bump function parameter h0i in (7) can be designedas follows:

h0i ¼ro0 � rsi

ro0 � rzið8Þ

where rsi 2 R denotes the radius of the UAV sensing zone.

3.3. Obstacle function

To ensure that no UAV contacts the stationary obstacles that have been predetected in thecombat zone, a scalar obstacle function is introduced to determine the relationship between theUAVs and the obstacles.

Definition 3The ith obstacle function bið�Þ 2 R; i ¼ 1; 2; . . . ; no; is a function satisfying

bi ¼0

At least one UAV contacts

the ith obstacle

> 0 Otherwise

8>><>>: ð9Þ

where no 2 R is the number of the obstacles.

As indicated by (9), if any of the UAVs approach the boundary of the ith obstacle, the ithobstacle function bið�Þ approaches zero, and hence, this type of design for bið�Þ spawns agradient-like term to ensure that the UAV moves away from the ith obstacle. Based on (9), theith obstacle function bið�Þ can be designed as follows:

bi ¼Ynj¼1

½jjqj � qoijj2 � ðrzj þ roiÞ2� ð10Þ

where qoi 2 R3; i ¼ 1; 2; . . . ; no; denotes the position of ith obstacle, and roi 2 R; i ¼ 1; 2; . . . ; no;denotes the radius of the ith obstacle.

3.4. EA detection zone function

To avoid being detected by the EAs, a scalar EA detection zone function is introduced todetermine the relationship between the UAVs and the EAs.

J. CHEN ET AL.120


DOI: 10.1002/acs

Definition 4The EA detection zone function gið�Þ 2 R; i ¼ 1; 2; . . . ; ne; is a function satisfying

gi ¼

0

At least one UAV reaches

the detection zone boundary

of the ith EA

1The ith EA is outside of the

sensing zone of all UAVs

2 ð0; 1Þ Otherwise

8>>>>>>>>>>>><>>>>>>>>>>>>:

ð11Þ

where ne 2 R is the number of the EAs.

As the boundary of the sensing zone of any of the UAVs approaches the boundary of thedetection zone of any of the EAs, the value of the EA detection zone function gið�Þ approacheszero. Based on the definition of the bump function in (4) and the EA detection zone function in(11), a specific EA detection zone function gið�Þ can be designed as follows:

gi ¼Ynj¼1

1� rhijhij

rdi þ rzjjjqj � qeijj

� �� ð12Þ

where qei 2 R3; i ¼ 1; 2; . . . ; ne; denotes the position of ith EA, and rdi 2 R3; i ¼ 1; 2; . . . ; ne;denotes the radius of the ith EA detection zone. From (11) and(12), it is clear that: (i) gi ¼ 0when for any j ¼ 1; 2; . . . ; n; jjqj � qei jj4rdi þ rzj ; and (ii) gi ¼ 1 when for all j ¼ 1; 2; . . . ; n;jjqj � qei jj5rsj : Based on (4), (11), and (12), the bump function parameter hij ; i ¼ 1; 2; . . . ; ne;j ¼ 1; 2; . . . ; n; in (12) can be designed as follows:

hij ¼ ðrdi þ rzjÞ=rsj ð13Þ

Remark 1Because gið�Þ is a user-defined function, and rdi may be somewhat uncertain, a best guess for rdican be used instead of the actual size of the EA detection zone as long as the best guest of rdi isgreater than the actual values.

3.5. UAV collision function

To avoid collisions between UAVs, we quantify the distance between the ith UAV and the jthUAV by introducing the UAV collision function.

Definition 5The UAV collision function bijð�Þ 2 R; i; j ¼ 1; 2; . . . ; n and i=j; is a function satisfying

bij ¼0 The ith UAV contacts the jth UAV

> 0 Otherwise

(ð14Þ



DOI: 10.1002/acs

From (14), it is clear that when the ith UAV and the jth UAV do not contact each other,bijðqiðtÞ; qjðtÞÞ > 0: As indicated by (14), the function bijð�Þ approaches zero if the ith UAVapproaches the jth UAV, and hence, this type of design for bijð�Þ spawns a gradient-like termto ensure that there is no contact between the ith UAV and the jth UAV. Based on the definitionof UAV relationship function in (14), a specific UAV collision function can be designed asfollows [26]:

bij ¼ jjqj � qi jj2 � ðrzi þ rzjÞ

2 ð15Þ

4. MULTIPLE UAV COOPERATIVE CONTROL

4.1. Multiple UAV navigation function

The primary objective is to navigate the multiple UAVs from an initial configuration to aconstant goal configuration while remaining in the combat zone and avoiding stationaryobstacles and EAs. To avoid collision with obstacles and EAs, the UAV configurationqðtÞ should remain in a free configuration space denoted by D� R3n which is a subset of thewhole combat zone with all configurations removed that involve collision with obstacles andEAs. The constant initial and goal configuration are assumed to be in the interior of D: Togenerate qðtÞ 2 D; the special artificial potential function coined a navigation function inReference [18], can be used. Specifically, the navigation function used in this paper is defined asfollows [10].

Definition 6Let D be a compact connected analytic manifold with a boundary, and let qn be a goal point inthe interior of D: A mapping jðqÞ : D! ½0; 1�; is a navigation function if

1. It is analytic on D (at least the first and second partial derivatives exist and are boundedon D);

2. It has a unique minimum at qn;3. It obtains a maximum value on the boundary of D (i.e. admissible on D);4. It is a Morse function (i.e. the matrix of second partial derivatives, the Hessian, evaluated

at its critical points is non-singular).

After defining the obstacle functions, the UAV collision functions, the boundary functions,and the EA detection zone function, we now introduce the navigation function that is utilized toderive the cooperative control algorithm. The navigation function jðqÞ 2 R for multiple UAVs isconstructed as follows:

jðqÞ ¼jjq� qnjj2

½jjq� qnjj2k þ GðqÞ�1=kð16Þ

J. CHEN ET AL.122


DOI: 10.1002/acs

where k 2 Rþ is a positive constant parameter, GðqÞ ¼4 G1G2G3 2 R; and the scalar functionsG1ð�Þ;G2ð�Þ;G3ð�Þ 2 R are defined as follows:

G1 ¼Yn0i¼0

biðqÞ G2 ¼Ynei¼1

giðqÞ ð17Þ

G3 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiYn

i¼1

Yj=i

bijðqÞr

where bið�Þ; gið�Þ; and bijð�Þ were defined in (7), (10), (12), and (15). Based on (17) and thedefinition of Gð�Þ; we can easily show that G ¼ 0 will occur:

* if G1 ¼ 0 (when any UAV contacts the boundary of the combat zone or the obstacles);* if G2 ¼ 0 (when any UAV contacts the detection zone boundary of a EA);* if G3 ¼ 0 (when two or more UAVs contact).

Remark 2Indeed, for a typical obstacle avoidance problem, it does not seem possible to construct jðqðtÞÞ;coined a navigation function, such that ð@=@qÞjðqðtÞÞ ¼ 0 only at qðtÞ ¼ qn: Koditschek [9] hasshown that strict global navigation is not possible. That is, as discussed in Reference [10], theappearance of interior saddle points (i.e. unstable equilibria) seems to be unavoidable; however,these unstable equilibria do not really cause any difficulty in practice. As proved in Reference[9], the function jð�Þ is a navigation function, so long as the parameter k exceeds a certainfunction of the geometric data. That is, jðqðtÞÞ can be constructed such that the attractiondomain of the unstable equilibria is a set of measure zero.

Remark 3The boundary of D includes the boundary of the combat zone, the boundary of the stationaryobstacles, and the detection zone boundary of the EAs.

Remark 4The main difference between the navigation function in (16) and the standard navigationfunction in Reference [10] is that for the navigation function in (16), the boundary of the combatzone and the EAs have only local interaction with the UAVs. When the boundary of the combatzone and EAs are out of the sensing zone of the UAVs, the corresponding boundary functionand EA detection zone function become unity. When the boundary of the combat zone contactsat least one UAV and the detection zone boundary of the EA contacts at least one UAV, theboundary function and corresponding EA detection zone function become zero. It is theanalytical smoothing bump function that allows the functions given by (7) and (12) to beappropriately smoothed out among all cases to facilitate the gradient-like differentiation formultiple UAV navigation. With a similar structure as compared to the standard navigationfunction given in Reference [9], it is not difficult to follow the proof in Reference [9] to show themultiple UAV navigation function defined in (16) satisfies the navigation function propertiesdefined in References [9, 10].



DOI: 10.1002/acs

4.2. Cooperative control design

The cooperative control in this paper is specified as all UAVs act based on all other UAVs’position to avoid collision with other UAVs, obstacles, and EAs. Based on the above definitions,the input uðtÞ in (2) is designed as follows:

u ¼ �K@j@q

� �T

ð18Þ

where K 2 R3n�3n is a constant positive definite symmetric control gain matrix andð@j=@qÞðqÞ 2 R1�3n is the partial derivative of jð�Þ from (16), along qðtÞ: The control algorithmdesigned in (18) illustrates this cooperative control concept since the multiple UAV navigationfunction jðqÞ is a function of all of the UAVs’ position.

Remark 5The control uðtÞ in (18) has been based on the kinematic model in (2). This control canbe extended using standard backstepping techniques to include second-order dynamicrobot models.

Remark 6With the definition of the EA zone detection function, the navigation function jðqÞ is a functionof ith EA position only when the ith EA is within the sensing zone of UAVs. Therefore, thecontrol designed in (18) depends on the position of the ith EA only when the ith EA is within thesensing zone of UAVs in which case the position of the ith EA is known. This is one of thenovelties of the proposed control algorithm.

4.3. Stability analysis

Theorem 1Provided qð0Þ 2 D; the cooperative control designed in (18) along with the navigation functionjðqÞ designed in (16) ensures that qðtÞ 2 D and asymptotic navigation in the sense that

qðtÞ ! qn as t!1 ð19Þ

ProofLet VðqÞ : D! R denote the following non-negative function:

VðqÞ ¼ jðqÞ ð20Þ

After taking the time derivative of (20), the following expression can be obtained:

’V ¼@j@q

’q ð21Þ

After substituting (18) into (21), the following expression is obtained:

’V ¼ �f ðtÞ ð22Þ

J. CHEN ET AL.124


DOI: 10.1002/acs

where f ðtÞ 2 R denotes the following non-negative function:

f ðtÞ ¼4@j@q

K@j@q

� �T

ð23Þ

Based on (22) and (23), it is clear that VðqðtÞÞ is a non-increasing function in the sense that

VðqðtÞÞ4Vðqð0ÞÞ ð24Þ

From (24), it is clear that for any initial condition qð0Þ 2 D; that qðtÞ 2 D 8t > 0; therefore,@j=@q; @2j=@q2 2L1 based on the Property 1 of Definition 6. Then, it is clear from (2) and (23)that ’f ðtÞ 2L1: Based on (20), (22), (23), and the fact that ’f ðtÞ 2L1 on D; then Lemma A.6 ofReference [27] can be invoked to prove that

@j@q

��

��! 0 ð25Þ

in the region D: Based on Definition 6 and Remark 2, it can be determined that if@j=@qðqðtÞÞ ! 0 then qðtÞ ! qn: &

Remark 7Based on Assumption 7 and (10), (12), (15), and (17), G1ðqð0ÞÞ; G2ðqð0ÞÞ; G3ðqð0ÞÞ > 0: ThereforeGðqð0ÞÞ > 0: Since Gðqð0ÞÞ > 0 and VðqðtÞÞ4Vðqð0ÞÞ for all time; then it is clear from (16) thatGðqðtÞÞ > 0: Then G1ðqðtÞÞ=0;G2ðqðtÞÞ=0;G3ðqðtÞÞ=0: Therefore G1ðqðtÞÞ;G2ðqðtÞÞ;G3ðqðtÞÞ > 0:We can now see that if GðqðtÞÞ > 0 for all time and qðtÞ ! qn as t!1; then all the objectivesgiven in the Problem Formulation Section will be achieved.

5. MOVING EA EXTENSION

As explained in the previous sections, the above navigation problem involves moving multipleUAVs from an initial configuration to a constant goal configuration while avoiding stationaryobstacles and EAs. To address the moving EA problem, we now discuss, in a heuristic manner,how the approach can still be expected to provide reasonable performance with regard tocooperative control.

A composite configuration of the moving EAs, denoted by qeðtÞ 2 R3ne ; is defined as follows:

qeðtÞ ¼ ½qTe1ðtÞ qTe2ðtÞ � � � q

TeneðtÞ�T ð26Þ

Similar to the navigation function jðqÞ designed in (16), a navigation-like functionj1ðq; qeÞ : D�D![0, 1] is constructed as follows:

j1ðq; qeÞ ¼jjq� qnjj2

½jjq� qnjj2k þ Gðq; qeÞ�1=kð27Þ

where Gðq; qeÞ ¼4G1G2G3 2 R; and the scalar functions G1ð�Þ;G3ð�Þ 2 R were defined in (17), and

G2ð�Þ 2 R are defined as follows:

G2 ¼Ynei¼1

giðq; qeÞ ð28Þ



DOI: 10.1002/acs

In (28), gið�Þ were defined in (12). To facilitate the subsequent discussion, the followingassumption must be satisfied:

Assumption 8The velocity of the EA must satisfy jjð@j1=@qeÞ’qejj4e where e 2 R is a positive constant.

Similar to the input uðtÞ designed in (18) for stationary EA case, the input uðtÞ in (2) formoving EA case is designed as follows:

u ¼ �K1@j1

@q

� �T

ð29Þ

where K1 2 R3n�3n is a constant positive definite symmetric control gain matrix.To discuss the stability and convergence of multiple UAVs system for the moving EA case, a

non-negative function V1ðq; qeÞ : D�D! ½0; 1� is defined as follows:

V1ðq; qeÞ ¼ j1ðq; qeÞ ð30Þ

After taking the time derivative of (30), the following expression can be obtained:

’V1 ¼@j1

@q’qþ

@j1

@qe’qe ð31Þ

Based on (2), (29), and Assumption 8, ’V1ðtÞ in (31) can be upper bounded as follows:

’V14� k@j1

@q

��

��2

þ e ð32Þ

where k¼4 lminðK1Þ; and lminð�Þ denotes the minimum eigenvalue. We now discuss two cases.

Case 1: qðtÞ is near the boundary of D:Since we are near the boundary, it seems reasonable to assume that jj@j1=@qjj5e; where e 2 R

is a positive constant. Hence, based on (32), k can be selected to ensure that ’V1ðtÞ be non-positive when qðtÞ is near the boundary of D: Similar to Property 3 in Definition 6, it is clearfrom (27) that the navigation-like function j1ðq; qeÞ obtains the maximum value on theboundary of D: Since j1ðtÞ is a non-increasing function, and j1ðtÞ51 for all time when qðtÞ isnear the boundary of D; the UAVs will never contact the boundary of D: It is also clear thatGiðtÞ > 0; for i ¼ 1; 2; 3 when qðtÞ is near the boundary of D: Hence, Objective 1–Objective 4 willbe achieved for this case.

Case 2: qðtÞ is far away from the boundary of D:For this case, the second term in (31) will vanish based on the property of the bump function

in (6). Hence, an argument identical to the proof of Theorem 1 can now be applied to show thatObjectives 1–5 will be achieved.

6. SIMULATION RESULTS

To illustrate the performance of the proposed cooperative control algorithm, we set up a2-dimensional simulation to navigate five UAVs from an initial configuration to the goalconfiguration while avoiding four obstacles and avoiding being detected by three EAs.Whenever there are multiple obstacles along the trajectory between the initial and final points,

J. CHEN ET AL.126


DOI: 10.1002/acs

which in this case consists of stationary EAs and obstacles, the UAVs cannot pass through theobstacles. The chosen initial configurations constitute non-trivial setups since the straight pathsconnecting the initial and final positions are obstructed by the stationary EAs and obstacles. Asa result, the UAVs might split along the straight line connecting the two points in two or smallergroups while avoiding the stationary EAs and obstacles. The downloadable video clips formultiple UAV navigation for both stationary EAs and moving EAs are provided in [28].

The initial configuration of the five UAVs was selected as follows:

q1ð0Þ ¼4 ½0 � 20�T; q2ð0Þ ¼

4 ½3 � 20�T; q3ð0Þ ¼4 ½�3� 20�T

q4ð0Þ ¼4 ½�3 � 17�T; q5ð0Þ ¼

4 ½3 � 17�T

Their destination configuration was selected as follows:

qn1 ¼4 ½0 30�T; qn2 ¼

4 ½3 27�T; qn3 ¼4 ½�3 27�T; qn4 ¼

4 ½�3 22�T; qn5 ¼4 ½3 22�T

The stationary obstacles were positioned in the combat zone at

qo1¼4 ½�10 � 5�T; qo2¼

4 ½2 12�T; qo3¼4 ½�20 10�T; qo4¼

4 ½10 0�T

Three stationary EAs were located at

qe1¼4 ½�7 2:5�T; qe2¼

4 ½15 13�T; qe3¼4 ½0 � 10�T

The radii of the UAVs were selected as follows

rz1 ¼ 1; rz2 ¼ 1; rz3 ¼ 1; rz4 ¼ 1; rz5 ¼ 1

The radii of the combat zone and the obstacles were selected as follows:

ro0 ¼ 35; ro1 ¼ 2; ro2 ¼ 3; ro3 ¼ 3; ro4 ¼ 1

0 10 20 30 40

0

10

20

30

Figure 3. First stage.



DOI: 10.1002/acs

The radii of the UAV sensing zones were selected as follows:

rs1 ¼ 8; rs2 ¼ 8; rs3 ¼ 8; rs4 ¼ 8; rs5 ¼ 8

The radii of the EA detection zone were selected as follows:

rd1 ¼ 4; rd2 ¼ 4; rd3 ¼ 4

0 10 20 30 40

0

10

20

30

Figure 4. Second stage.

0 10 20 30 40

0

10

20

30

Figure 5. Third stage.

J. CHEN ET AL.128


DOI: 10.1002/acs

Figures 3–7 show several stages of multiple UAVs from the initial configuration to the goalconfiguration. The grey circles illustrate the detection zone of the EAs, the black circles illustratethe obstacles, and the UAVs are denoted by the circles with the cross in the centre. The fivecrosses at the top part of the figure illustrate the goal configuration of the UAVs. From the

0 10 20 30 40

0

10

20

30

Figure 6. Fourth stage.

0 10 20 30 40

0

10

20

30

Figure 7. Goal configuration.



DOI: 10.1002/acs

figures, it is clear that the UAVs moved to their destination while avoiding the stationaryobstacles and avoiding being detected by the EAs along their paths.

7. CONCLUSION

A navigation function based cooperative control is developed in this paper to navigate multipleUAVs in an obstacle and enemy asset cluttered environment. The standard navigation functionapproach was extended to a multiple navigation strategy with analytical switching amongdifferent cases due to the limited sensing zone of the UAVs. A differentiable cooperative controllaw is proposed based on this navigation function that yields asymptotic convergence. Thenavigation strategy of avoiding the moving EAs is also discussed. Simulation results areprovided to demonstrate the performance of the proposed control strategy.

REFERENCES

1. Latombe JC. Robot Motion Planning. Kluwer Academic Publishers: Boston, MA, 1991.2. Khatib O. Commande dynamique dans l’espace operational des robots manipulateurs en presence d’obstacles. Ph.D.

Dissertation, Ecole Nationale Supeieure de l’Aeeronatique et de l’Espace (ENSAE), France, 1980.3. Barraquand J, Langlois B, Latombe JC. Numerical potential fields techniques for robot path planning. IEEE

Transactions on Systems, Man, and Cybernetics 1992; 22:224–241.4. Barraquand J, Latombe JC. A Monte-Carlo algorithm for path planning with many degrees of freedom. Proceedings

of the IEEE International Conference on Robotics and Automation, Cincinnati, OH, May 1990; 584–589.5. Connolly CI, Burns JB, Weiss R. Path planning using laplace’s equation. Proceedings of the IEEE International

Conference on Robotics and Automation, Cincinnati, OH, May 1990; 2102–2106.6. Khatib O. Real-time obstacle avoidance for manipulators and mobile robots. International Journal of Robotics

Research 1986; 5(1):90–99.7. Volpe R, Khosla P. Artificial potential with elliptical isopotential contours for obstacle avoidance. Proceedings of the

IEEE Conference on Decision and Control, Los Angeles, CA, December 1987; 180–185.8. Koditschek DE. Exact robot navigation by means of potential functions: some topological considerations.

Proceedings of the IEEE International Conference on Robotics and Automation, Raleigh, NC, May 1987; 1–6.9. Koditschek DE, Rimon E. Robot navigation functions on manifolds with boundary. Advances in Applied

Mathematics 1990; 11:412–442.10. Rimon E, Koditschek DE. Exact robot navigation using artificial potential function. IEEE Transactions on Robotics

and Automation 1992; 8(5):501–518.11. Bemporad A, De Luca A, Oriolo G. Local incremental planning for a car-like robot navigating among obstacles.

Proceedings of the IEEE International Conference on Robotics and Automation, Minneapolis, MN, April 1996;1205–1211.

12. Ge SS, Cui YJ. New potential functions for mobile robot path planning. IEEE Transactions on Robotics andAutomation 2000; 16(5):615–620.

13. Guldner J, Utkin VI, Hashimoto H, Harashima F. Tracking gradients of artificial potential field with non-holonomic mobile robots. Proceedings of the American Control Conference, Seattle, WA, June 1995; 2803–2804.

14. Kyriakopoulos KJ, Tanner HG, Krikelis NJ. Navigation of nonholonomic vehicles in complex environments withpotential fields and tracking. International Journal of Intelligent Control of Systems 1996; 1(4):487–495.

15. Laumond JP, Jacobs PE, Taix M, Murray RM. A motion planner for nonholonomic mobile robots. IEEETransactions on Robotics and Automation 1994; 10(5):577–593.

16. Tanner HG, Kyriakopoulos KJ. Nonholonomic motion planning for mobile manipulators. Proceedings of the IEEEInternational Conference on Robotics and Automation, San Francisco, CA, April 2000; 1233–1238.

17. Tanner HG, Loizou SG, Kyriakopoulos KJ. Nonholonomic navigation and control of cooperating mobilemanipulators. IEEE Transactions on Robotics and Automation 2003; 19(1):53–64.

18. Loizou SG, Kyriakopoulos KJ. Closed loop navigation for multiple holonomic vehicles. Proceedings of the IEEE/RSJ International Conference on Intelligent Robots and Systems, Lausanne, Switzerland, October 2002; 2861–2866.

19. Loizou SG, Tanner HG, Kumar V, Kyriakopoulos KJ. Closed loop motion planning and control for mobile robotsin uncertain environments. Proceedings of the 42nd Conference on Decision and Control, Maui, HI, U.S.A.,December 2003; 2926–2931.

J. CHEN ET AL.130


DOI: 10.1002/acs

20. Chang DE, Shadden S, Marsden J, Olfati-Saber R. Collision avoidance for multiple agent systems. Proceedings ofthe 42nd IEEE Conference on Decision and Control, Maui, HI, U.S.A., December 2003; 539–543.

21. Dimarogonas DV, Zavlanos MM, Loizou SG, Kyriakopoulos KJ. Decentralized motion control of multipleholonomic agents under input constraints. Proceedings of the 42nd IEEE Conference on Decision and Control, Maui,HI, U.S.A., December 2003; 3390–3395.

22. Tanner HG, Jadbabaie A, Pappas GJ. Stability of flocking motion. Technical Report No. MS-CIS-03-03,Department of Computer and Information Science, University of Pennsylvania, January 2003.

23. Tanner HG. Flocking with obstacle avoidance in switching networks of interconnected vehicles. Proceedings of theIEEE International Conference on Robotics and Automation, New Orleans, LA, April 2004; 3006–3011.

24. Olfati-Saber R. Flocking for multi-agent dynamic systems: algorithms and theory. IEEE Transactions on AutomaticControl 2006; 51(3):401–420.

25. Saber RO, Murray RM. Flocking with obstacle avoidance: cooperation with limited communication in mobilenetworks. Proceedings of the 42nd IEEE Conference on Decision and Control, Maui, HI, U.S.A., December 2003;2022–2028.

26. Whitcomb LL, Koditschek DE. Automatic assembly planning and control via potential functions. Proceedings of theIEEE/RSJ International Workshop on Intelligent Robots and Systems, Osaka, Japan, November 1991; 17–23.

27. de Queiroz MS, Dawson DM, Nagarkatti SP, Zhang F. Lyapunov-Based Control of Mechanical Systems.Birkhauser: Basel, 1999.

28. Chen J, Dawson DM, Salah M, Pradhan N. Multiple UAV navigation with finite sensor range. Clemson UniversityCRB Technical Report, CU/CRB/3/3/05/#2, http://www.ces.clemson.edu/ece/crb/publictn/tr.htm, March 2005.



DOI: 10.1002/acs

Cooperative control of multiple vehicles with limited sensing

Documents

Transcript of Cooperative control of multiple vehicles with limited sensing