In this paper, we propose a control law to achieve a rendezvous of autonomous vehicles moving in three-dimensional (3D) space, usingminimal data sensing and quantized control. A pre-assigned graph uniquelyassigns the pursuer-target pair in a cyclic manner. A quantized control law has been proposed whichallows the vehicle to pitch and yaw simultaneously in the required direction and track its target agent.The onlymeasurement required for the proposed control law is the quadrant fromwhich the target vehiclemoves out of the field-of-view of the pursuing vehicle. A Lyapunov function is chosen to find a domain forthe field-of-view which guarantees rendezvous under the proposed control law. Computer simulationsare presented to demonstrate the control law.
Autonomous vehicle systems have found potential applicationsin various military and civil operations. Greater benefits arise fromcooperation of a team of vehicles than individual. A multi-agentsystem is robust to failure andmore efficient than individual agentsin certain cases. It is also possible to reduce the size and operationalcost of individual agents and increase system reliability. This hasaroused interest in the control community in cooperative controland consensus algorithms. In Ren, Beard, and Atkins (2005), theauthors have mentioned various consensus algorithms in multi-agent coordination.
The rendezvous problem is one of the various consensusproblems where all agents of a multi-agent system converge to apoint at the same time. Rendezvous of autonomous agents usingvarious decentralized or distributed controls has been pursuedactively in the past few decades. Some papers which discusscontrollers for rendezvous are Ando, Oasa, Suzuki, and Yamashita(1999), Hui (2011) and Lin, Morse, and Anderson (2007a,b).The cyclic pursuit problem is closely related to the rendezvousproblem. There is a vast literature on control for cyclic pursuit,
✩ The material in this paper was partially presented at the 50th IEEE Conferenceon Decision and Control (CDC 2011) and European Control Conference (ECC), Dec.12–15, 2011, Orlando, Florida, USA. This paper was recommended for publicationin revised form by Associate Editor Jun-ichi Imura under the direction of EditorToshiharu Sugie.
some of which are Bruckstein, Cohen, and Efrat (1991), Galloway,Justh, and Krishnaprasad (2009, 2010), Marshall, Broucke, andFrancis (2004a,b), Pavone and Frazzoli (2007); Ramirez, Pavone,and Frazzoli (2009), Richardson (2001) and Sinha and Ghose(2006). The pursuit curves have been studied in Bernhart (1959). Inthe references cited above, information regarding relative position,angle or velocity is required to achieve the objective. Sometimestasks have to be donewithminimal data and quantized controllers.We mention some of the papers which look into minimalism andquantized control applications in the following parts.
Minimalismmeans, given anobjective to be achievedby a groupof autonomous agents, what is the minimum information neededto achieve the objective? Optimal navigation, pursuit–evasion,robot localization with minimal data have been discussed inFredslund and Matarić (2002), Sachs, LaValle, and Rajko (2004);Tovar, Guilamo, and LaValle (2005); Tovar, LaValle, and Murrieta(2003); Tovar,Murrieta-Cid, and LaValle (2007). Various sensorlessmanipulation tasks have been explored in Böhringer, Brown,Donald, Jennings, and Rus (1997). The use of quantized controland coarse quantized measurement of plant outputs (states) ismotivated by restricted information flow between plant andcontroller for various reasons. While the use of quantizedmeasurements for stabilization has been discussed in Brockett andLiberzon (2000) and Delchamps (1990); Ishii and Francis (2002),Nair and Evans (2000) and Petersen and Savkin (2001) discuss thecontrollers with a finite data rate communication link. In Elia andMitter (2001), the authors have presented the coarsest, least densequantizers for state-feedback controller and estimator to stabilizea single-input–single-output linear time-invariant system.
The above cited references motivated us to look into theapplication of minimalism and quantized control to a multi-agent
520 S.R. Sahoo et al. / Automatica 49 (2013) 519–525
Fig. 1. Schematic of vehicle and transformation from body-fixed frame to earth-fixed frame.
system. In Yu, LaValle, and Liberzon (2008), the authors haveproposed a quantized controller with minimal sensor data toachieve rendezvous of agents moving on a plane. However, mostautonomous agents, like unmanned aerial vehicles (UAVs) andautonomous underwater vehicles (AUVs), more often move in 3Dspace. Though the work presented here is on the lines of Yu et al.(2008), the 3D problem has significant differences and complexity,in the sense that the geometry in 3D is more involved than the2D case. Also, the extension of the notion of minimal sensing andcoarse actuation to the 3D case is non-trivial.
The paper is organized as follows: Section 2 presents the vehiclemodel, the sensors and the control law. Section 3 presents thecondition for rendezvous. Section 4 presents the simulation resultsfor the problem. Section 5 concludes the paper.
2. Problem formulation
We assume n agents modeled similar to a UAV with simplekinematics. The agent can yaw and pitch with constant angularspeed, and move in a straight path (cruise) with constant speed.The agent, however, cannot roll or slip laterally. Each agent hasa conical field-of-view with infinite range within which it triesto maintain its target. The target for the ith agent is the agent(i+1)modulo n. The control is applied only when the targetmovesout of the windshield. We assume that all the agents have theirtarget within their windshield initially. The vehicle model, sensorsand control are now presented.Vehicle model: Let pi = (xi, yi, zi) ∈ R3 be the position ofthe ith agent in an earth-fixed frame and (αi, βi, γi) be the Eulerangles corresponding to the Z–Y–Z Euler angle convention fortransformation from the body-fixed frame to the earth-fixed framewith anti-clockwise rotations. See Fig. 1. The forward (linear)velocity of the vehicle is only along its body Xb axis and itsmagnitude (vi) remains constant. The vehicle can rotate aboutits body Yb-axis (pitch) and about its body Zb-axis (yaw) in bothclockwise and counter-clockwise directions.
The pitch and yaw rates take values from the discrete set,ωyib∈
{−ωi, 0,+ωi} and ωzib∈ {−ωi, 0,+ωi}, where ωi is constant and
ωi > 0. The agents are considered identical and hence vi = vj andωi = ωj.
Since the model involves rigid body motion (translation androtation) in space, we relate the inertial velocity components tothe body velocity components through the Euler angles as
[xi yi zi]T = Rzi3Ryi2
Rzi1[vi 0 0]T (1)
where (xi, yi, zi) are the velocity components in the inertial frameand (vi, 0, 0) are the velocity components in the body-fixed frame.Rzi3, Ryi2
, Rzi1are rotation matrices, locally parametrized by the
Euler angles. The Euler angle rates are related to the vehicle bodyangular velocities as
[αi βi γi]T
= C(αi, βi, γi)[0 ωyibωzib
]T (2)
where C(αi, βi, γi) is thematrix corresponding to the angular ratesfor the Euler transformation. Eqs. (1) and (2) describe the kinematicmodel of the ith vehicle.Sensors: The sensor for each agent has a conical view with the halfangle of the cone being φ ∈ (0, π) as shown in Fig. 2(a). This field-of-view is termed as the windshield. The range of view within thisangle is assumed to be infinite. It is also assumed that one agentcannot occlude the view of another agentwhen both appearwithinthe windshield. The field-of-view, as seen by the agent, appearslike a disc. This disc can be divided into four quadrants as shownin Fig. 2(b). The sensors do not give the actual distance betweenagents. They only give a discrete output based on the quadrantfrom which the assigned agent (the target) moves out.
Let the output set be O and the sensor measurement of theith vehicle be (oyi , ozi) ∈ O. (oyi , ozi) takes the following valuesaccording to the quadrant from which the target agent j escapesfrom view:-
(oyi , ozi) =
(−sgn(Pyb), sgn(Pzb)), j escapes from P(0, 0), agent j is in the view (3)
in which agent j is the target agent of i and P is the point of escapefrom the windshield. These values are also indicated in Fig. 2(b).Controls: The output of the sensors actuates the controllers fornecessary action. The control law is defined with respect to thesensor output as:(ωyib
, ωzib) = (ozi , oyi)ωi. (4)
As can be seen, this control law involves no history or stateestimation. Next we try to find the conditions such that using thiscontrol law the agents achieve rendezvous.
3. Conditions for rendezvous
3.1. Concepts from graph theory
The agents are considered to be in cyclic pursuit. For simplicityand without loss of generality we assume that (i + 1)modulo n isassigned to agent i. This system is represented by a digraph G =
(V , E ) with the agents being its nodes (i ∈ V ) and ei,i+1 ∈ E (G ).As agent i catches upwith agent i+1, i and i+1move as one entityi + 1. This is called merging. The merging operation is triggeredwhen li,i+1 ≤ ρ. The merging radius, ρ(> 0), is the distancebetween the pursued and the pursuer after which they merge andmove as one entity. Unlike many studies on this problem whichconsider a pointmass geometry for the vehicles, we assume a finitegeometry and hence a safe zone is considered surrounding eachvehicle, which mathematically translates to the merging radius ρ.Merging occurs if ei,i+1 ∈ E (G ), li,i+1 ≤ ρ. After merging, agenti−1 starts pursuing i+1. The node i is deleted and the edges ei−1,iand ei,i+1 are deleted from E (G ) and a new edge ei−1,i+1 comes intoeffect. The number of nodes is reduced. The graph G is said to belive if it has at least one edge (Yu et al., 2008). A graph with a singlevertex is also a live-graph. If G is not live then there exists morethan one vertex with no edge.
3.2. A Lyapunov candidate
Let V : R3n→ R be a Lyapunov function which is defined as
V =
ei,j∈E (G )
li,j. (5)
Since V is the sum of distances, it will always be positive and willgo to zero only when the system achieves rendezvous. So V istermed as rendezvous positive definite. At each instant of mergingV has a discontinuity. V is also called a graph compatible Lyapunovfunction as it is based on the digraphG (Yu et al., 2008).G is stronglyconnected and hence G is live for t ≥ 0, and V is rendezvouspositive definite (from Lemmas 1 and 2 Yu et al., 2008).
S.R. Sahoo et al. / Automatica 49 (2013) 519–525 521
(a) The conical field-of-view ofthe agent.
(b) Field-of-view is a disc when seenfrom agent.
Fig. 2. The sensor of a vehicle.
Fig. 3. Two consecutive agents in cyclic pursuit.
3.3. Condition on the windshield angle
Refer to Fig. 3. Note that li−1,i, vi and li,i+1 need not be coplanar.
• pi−1, pi, pi+1 are the positions of agents i − 1, i and i + 1respectively measured in the earth-fixed frame.
• li−1,i, li,i+1 are the line-of-sight distances of agent i − 1 and irespectively.
• φi is the angle between the velocity vector vi and li,i+1.• θi is the smaller angle between li−1,i and li,i+1.• ψi is the smaller angle between vi and li−1,i.
All angles considered are positive and each agent has unit speed.Thus li−1,i = − cos(ψi)− cos(φi−1) and hence,
V = −
ni=1
(cos(φi)+ cos(ψi)). (6)
A necessary condition for rendezvous to occur is that V is strictlyless than zero. Note that V is discontinuous at the instants ofmerging. Let the instants whenmerging occurs be tk, k = 1, 2, . . . .These instants are called switching instants. As the number ofagents in the system is finite, the switching sequence {tk} is alsofinite. For an n-agent system, the maximum number of switchingsthat are possible is n− 1. However, since V is continuous betweentwo consecutive instants of merging [tk, tk+1), V is a piecewisecontinuous function. Hence, for V to be monotonically decreasing
• The switching sequence {tk} should be finite.• V (t) has to be strictly less than zero in the interval t ∈ [tk, tk+1).• V (tk+1 − ε) ≥ V (tk+1 + ε), 0 < ε ≪ 1.
Theorem 1. Unit speed cyclic pursuit of n agents with kinematicsgiven by (1) and (2) will rendezvous if the agents maintain their
targets within the windshield and the windshield angle φ is boundedas
0 < φ <
π/2 n = 2
minπ/n, cos−1
n − 1n
n ≥ 3. (7)
Proof. The proof of this theorem is divided into four parts. Firstwe prove V (tk+1 − ε) ≥ V (tk+1 + ε), 0 < ε ≪ 1. Next within[tk, tk+1) we rule out the condition φ ∈ [π/n, π]. Next we find acondition onφ for n = 2. Thenwe find the condition onφ for n ≥ 3and complete the proof.(1) V decreases after each merging: Let tk+1 be the instant whenagents i and i + 1 merge. Consider an interval [tk+1 − ε, tk+1 +
ε], 0 < ε ≪ 1. At tk+1−ε, V (tk+1−ε) =n
j=1,(j=i,i−1) lj,j+1(tk+1−
ε) + li−1,i(tk+1 − ε) + li,i+1(tk+1 − ε). At tk+1 + ε, V (tk+1 +
li−1,i+1(tk+1−ε). Now limε→0+ [li−1,i(tk+1−ε)+ li,i+1(tk+1−ε)] =
li−1,i(tk+1) + ρ > li−1,i(tk+1) = limε→0+ li−1,i+1(tk+1 + ε). Hencewe have, li−1,i(tk+1 − ε) + li,i+1(tk+1 − ε) > li−1,i+1(tk+1 + ε).Hence, V (tk+1 + ε) < V (tk+1 − ε). Thus, V decreases when theagents merge (see Fig. 4).(2) Rule out φ ∈ [π/n, π]:
Lemma 3.1. For any integer n ≥ 2, the windshield angle φ = π/npermits trajectories for which V = 0.
Proof. The proof follows on the similar lines of Yu et al. (2008). �
When φ ∈ (π/n, π] for the same case as in the proof of Lemma 3.1with φi = φ =
πn + δ, where δ is a small positive angle we have
ψi = θi+φi = n−1
n
π+δ, ∀i. So, V > 0. Hence, whenφ ∈ [
πn , π]
we cannot ensure that we will always have V < 0.(3) Condition on φ for n = 2: Consider n = 2, then ψ1 = φ2 andψ2 = φ1. For V < 0, from (6), we have
− 2(cos(φ1)+ cos(φ2)) < 0. (8)
Eq. (8) will always holds if φ < π/2 imposing cos(φ1) > 0 andcos(φ2) > 0. Thus, for n = 2, when φ < π/2, V < 0 is alwayssatisfied.
(a) Just beforemerging.
(b) Just aftermerging.
Fig. 4. Merging of agent i and i + 1. New edge ei−1,i+1 is created. Edges ei−1,i and ei,i+1 are deleted.
522 S.R. Sahoo et al. / Automatica 49 (2013) 519–525
Fig. 5. Triangle inequality of angles.1ABC and1ADC are on different planes.
Fig. 6. A closed polygonal line with αi ’s and βi ’s.
(4) Condition on φ for n ≥ 3: To establish a relation betweenψi, θi and φi, we state a lemma partly similar to the one stated inBruckstein et al. (1991).
Lemma 3.2. Let ABC and ACD be two triangles inR3 with a commonside AC as shown in Fig. 5. Define the three angles α = BAC, β =
CAD, γ = BAD(0 ≤ α, β, γ ≤ π). Then
γ ≤ min{α + β, 2π − (α + β)} (9)
with equality holding only if A, B, C,D are coplanar.
Proof. The proof of this lemma is given in Appendix A. �
Consider the triangles formed by the points pi−1, pi, pi+1 and thevelocity vector vi as shown in Fig. 3. Applying Lemma 3.2 we have
ψi ≤ min{(θi + φi), (2π − (θi + φi))}. (10)
In Fig. 6, if βi is the angle formed by the polygonal line at the i-thvertex such that 0 < βi < π and αi be the supplementary angle ofβi then the following lemma holds.
Lemma 3.3 (Lemma 3, Bruckstein et al., 1991). In any closedpolygonal line with n segments in R3, the inequalities
ni=1 βi ≤
(n − 2)π andn
i=1 αi ≥ 2π , hold. Equality occurs iff the polygonalline is a planar convex polygon.
We now check for n ≥ 3. From Lemma 3.1 and the fact thatφi < φ we have,
0 < φi < φ < π/n. (11)
For any closed polygonal line we can always consider the smallerangle as an interior angle. So 0 < θi < π . Hence, we have
0 < θi + φi < π + π/n. (12)
For 0 ≤ ψi ≤ π and 0 < θi + φi ≤ π , from (10) we have− cos(ψi) ≤ − cos(θi + φi). Similarly, for 0 ≤ ψi ≤ π and π <θi +φi < π +π/n, from (10) we have,− cos(ψi) ≤ − cos(θi +φi).Thus we have, for all ψi ∈ [0, π], θi + φi ∈
0, π +
πn
− cosψi ≤ − cos(θi + φi). (13)
From (6) and (13),
V ≤ −
ni=1
(cosφi + cos(θi + φi)) =: V1. (14)
V1 can now be written as the sum of two functions, f and g , asfollows
f := −
ni=1
cos(θi + φi) (15)
g := −
ni=1
cosφi. (16)
Note that −n ≤ f , g ≤ n. From (11) we have g < 0. For V1 to benegative definite−g > f for all values of (θi+φi). f can be positiveor negative depending on the value of (θi+φi) for all i. For negativevalue of f , it is guaranteed that V1 < 0. So we need find conditionssuch that V1 < 0 evenwhen f > 0.We have the following possiblesets on which f can be analyzed.
(a) Θ+
1 = {(θ1, . . . , θn)|∀i, (θi + φi) ∈ (0, π2 )}. In this set f isalways negative. Hence, V1 < 0.
(b) Θ+= {(θ1, . . . , θn)|∃i, (θi + φi) ∈ (0, π2 )}. In this set f can
have positive values.(c) Θ−
1 = {(θ1, . . . , θn)|∃i, (θi + φi) ∈ [π2 , π +
πn )}. f can take
both positive and negative values in this set.(d) Θ−
= {(θ1, . . . , θn)|∀i, (θi + φi) ∈ [π2 , π +
πn )}. f is always
positive in this set.
Clearly Θ+
1 ⊂ Θ+ and Θ−⊂ Θ−
1 . Thus, to have V1 < 0 we haveto ensure that −g is always greater than the maximum value f canattain inΘ+
∪Θ−. Consider these two mutually disjoint sets thatsatisfy (12).
We find the maximum value that f can attain in each of the setsseparately. If (θ1, . . . , θn) ∈ Θ+ then for at least one i, (θi + φi)must belong to [0, π/2). Let θk+φk ∈ [0, π/2). So− cos(θk+φk) <0. Now,
f < −
n−1i=1,i=k
cos(θi + φi) < (n − 1). (19)
Now the behavior of f has to be analyzed in Θ−. Here, we statea lemma similar to the one stated in Yu et al. (2008), though theproof is significantly different.
Lemma 3.4. Unit speed cyclic pursuit of n agents satisfying (11) hasthe property that the function f (Θ,Φ) has a single stationary pointinΘ−.
Proof. Keeping φi’s constant, f only depends on θi’s. Let
h :=
ni=1
θi ≤ (n − 2)π. (20)
Converting (20) to equality we haven
i=1 θi − (n − 2)π = −β2,where β ∈ R is termed a slack variable. The constraint can be thuswritten as an equality H := h + β2
− (n − 2)π = 0. The equalityconstraint is now incorporated into the optimization problemwiththe help of a Lagrange multiplier λ as fmod(θ1, . . . , θn, β, λ) =
f − λH . Note that by converting the inequality constraint to anequality constraint we have introduced an additional variable βinto the problem. The stationarity conditions for all i are
sin(θi + φi) = λ, 2λβ = 0, H = 0. (21)
From (21) we have the following possibilities for β and λ − (β =
λ = 0), (β = 0, λ = 0), (β = 0, λ = 0). For the first 2solutions, when λ = 0 we have from (21), (θi + φi) = ±nπ,∀i,
S.R. Sahoo et al. / Automatica 49 (2013) 519–525 523
(a) Movement of windshieldto bring the target back intothe field-of-view.
(b) The dotted circle is the locusof vi+1 . The dashed circle is thelocus of vi,i+1 .
(c) Minimum ω.
Fig. 7. Condition on ω.
n = 0,±1,±2, . . . . Since we are analyzing f in Θ−, the values of(θi + φi)must belong to [π/2, π + π/n), for all i. So (θi + φi) = πand
ni=1
(θi + φi) = nπ. (22)
From (11) and (20), we have
0 <
(θi + φi) < (n − 1)π. (23)
It is clear that (22) violates (23). So, λ = 0 cannot be a solution.Hence, the first 2 solutions are ruled out. The only possible solutionis β = 0, λ = 0. As we are considering the set Θ−, (23) can bereduced tonπ2
≤
(θi + φi) < (n − 1)π. (24)
Now, if (θi+φi) ≥n−1n π, ∀ i then
ni=1(θi+φi) ≥ (n−1)π , which
clearly violates (24). For all i, (θi + φi) cannot be less than π2 as we
are considering the set Θ−. So to satisfy (24), there exists at leastone i (say k) such that (θ ck +φk) ∈ (π2 ,
n−1n π). So sin(θ ck +φk) > 0,
where (θ c1 , . . . , θcn ) is the stationary point. From (21), λ = sin(θ c1 +
φ1) = sin(θ c2 + φ2) = · · · = sin(θ ck + φk) = · · · = sin(θ cn + φn).We are considering the set Θ−. So, (θ c1 + φ1) = · · · = (θ cn + φn).Thus, f (Θ,Φ) has a single stationary point inΘ−. �
Lemma 3.5. The stationary point is a point of maxima inΘ− and
fmax < −n cos(n − 2)π + nφ
n
. (25)
Proof. The proof of this lemma is given in Appendix B. �
Lemma 3.6. For n agents in cyclic pursuit with unit speed, f (Θ,Φ)satisfies
fmax(Θ,Φ) < maxn − 1,−n cos
(n − 2)π + nφ
n
. (26)
Proof. f is a continuous function onΘ+∪Θ−. Themaximumvalue
that f can attain is the maximum of the two upper bounds in (19)and (25). Hence, (26) follows. �
In continuation of the proof of Theorem 1, for V1 < 0,−g > f .We have to ensure −g > max
n − 1, − n cos
(n−2)π+nφ
n
.
Hence, n cos(φ) > maxn − 1, − n cos
(n−2)π+nφ
n
which
implies φ < mincos−1
n−1n
, π
n
. Hence, for n ≥ 3, and V1
to be less than zero, (7) should be satisfied. From (14), V < V1, sofor the same range of φ, V < 0. Hence, proved. �
3.4. Condition on angular speed
Theorem 1 assumes that the agents always have their targetswithin their windshield. To achieve this, the angular speed of theagents need to satisfy some conditions.
Theorem 2. If the angular speed ω ≥πvρ, it is sufficient for each
agent to track its target.
Proof. Consider Fig. 7(a). Let O be the position of agent i with itsheading along OV and one of the boundaries of the field-of-viewcone of the agent is along OF. P is a point on OF from which agenti + 1 exits the field-of-view of agent i. Also OP ≥ ρ where ρis the merging radius. The relative velocity of agent i + 1 withrespect to i is vi,i+1 = vi+1 − vi. Consider Fig. 7(b). The locus ofvi,i+1 with P as its initial point is a sphere with P being a pointon the surface of the sphere. For a given ρ and v consider a smalltime interval 1t and an angular velocity ω′ such that the surfaceof sensor (field-of-view) cone is made tangent to the sphere (thesphere being brought just inside the field-of-view cone) and theboundaryOF ′ being perpendicular to the diameter AP of the spherepassing through P . The relative speed of agent (i+ 1)with respectto i along AP is 2v. Hence, AP = 2v1t . From the geometry shownin Fig. 7(a), we have AP = OP cos(φ) ≥ ρ cos(φ)which implies
1t ≥ρ
2vcos(φ). (27)
In the time interval 1t , the distance traveled by agent i + 1 ismaximum when it moves along AP as compared to the distancemoved by it in any other direction in the same time. In such ascenario, to bring back agent i + 1 into the field-of-view of agenti, the windshield boundary OP should coincide with OP ′ in a timeless than or equal to time interval1t . So
ω′1t ≥π
2− φ. (28)
To satisfy (28) and from (27), ω′≥
π2 −φ
cos(φ)
2vρ
a sufficiency
condition for which is ω′≥
πvρ. Consider Fig. 7(c). When agent
i + 1 moves out of the field-of-view of agent i from the point P ,agent i starts turning due to the control applied to bring back thetarget into field-of-view. The effective angular velocity that bringsthe target back into view is about z+-axis and it should be
ωz+ = ω′= ωy sin(δ)+ ωz cos(δ) (29)
where δ is the angle of rotation of the y− z axes to coincide with P .From (29), for δ ∈ (0, 2π) the minimum value of |ω′
| is ω. Hence,ω ≥
πvρ. �
Hence with a choice of finite angular speed satisfying Theorem 2the agent is able to turn at a sufficient fast rate and track its target.Theorems 1 and 2 give sufficiency conditions on the windshield
524 S.R. Sahoo et al. / Automatica 49 (2013) 519–525
(a) Trajectory of agents. ‘o’ indicates initial positions of the agents. (b) Distance (V ) and individual distance between consecutive agents.
(c) Value of total distance (V ) for different values of φ. (d) Value of V different values of ω.
Fig. 8. Simulation results for five agents.
angle φ and angular speed ω of each agent that guaranteerendezvous. It can be noted that as the number of agents in thesystem increase the upper bound on φ reduces and the choiceof the angle becomes restricted. However, the angular speed ω isindependent of the number of agents in the system.
4. Simulation results for rendezvous in 3D
We considered a ‘‘five-agent’’ system. The agents start fromrandom points in space with the respective target agent inside thefield-of-view. The forward speed for each agent is 10 units/s andρ = 0.15. The limits on φ and ω are π
5 and 209.4 respectively.Considering φ = 0.2π5 and ω = 210 rad/s we find the agentsconverge to a point. See Fig. 8(a) and (b).
4.1. Case A: Different values of φ
We carried out simulations for φ = 0.2π5 , 0.5π5 , 0.9
π5 , 1.02
π5 ,
1.3π5 with v = 10, ω = 210, ρ = 0.15. Fig. 8(c) shows acomparison between the sum of the distances as given in (5) fordifferent values of thewindshield angle. It is observed that the timefor rendezvous increases with increase in the windshield angle.This is because the control becomes coarse and the target-agentstays for a longer time in the field-of-view of the pursuing agent.
For φ = 1.02π/5 and φ = 1.3π/5, (7) is violated. It is seen thatforφ = 1.02π/5, the agents still rendezvouswhile forφ = 1.3π/5they do not. The condition on φ is a sufficient condition.
4.2. Case B: Different values of ω
We carried out simulations for ω = 2, 20, 210, 2000 whilekeeping v = 10, φ = 0.2π5 , ρ = 2. See Fig. 8(d). Here thecondition on φ is satisfied. It is observed that for ω > 209.4,Theorem 2 is satisfied and rendezvous is achieved. However, forω < 209.4, the agents do notmerge but they do not diverge either.
5. Summary and future scope of work
In the present work, agents modeled similar to a UAV withsimple kinematics have been considered. Each agent has a conicalfield-of-view and tries to maintain its target within its view. Wehave obtained sufficient conditions on the windshield angle andangular speed of the agents which ensure rendezvous of themulti-agent system. Our simulations support the bounds derived in thetheorems. Due to quantized control, chattering can be observed inthe control when the agent and its target are close to each other.The system evolution, however, does not suffer from chattering.
As an extension to the present work, one can determine a lowerbound on the time that rendezvous takes, predict the rendezvouspoint or achieve rendezvous at a specific location in space withsimilar sensing and controls. One can also look for simpler sensingand control law to achieve rendezvous. Investigating strategies forcollision avoidance between agents i and j, (j = i + 1), as alsolooking into other tasks like formation flying, flocking are otherpotential avenues for future work.
Acknowledgment
The authors acknowledge useful discussions with AnupMenon.
Appendix A. Proof of Lemma 3.2
Consider Fig. A.1. Without loss of generality consider AB =
AC = AD = r . Consider a sphere with center at A and radiusr . B,D, C lie on the surface of this sphere. 1BDC is a sphericaltriangle formed by the intersection of the great circles passingthrough BC, CD and DB. The angles subtended by
⌢
BC,⌢
CD and⌢
DBat the center are α, β and γ respectively. By triangle inequality⌢
DB ≤⌢
BC +⌢
CD and thus γ ≤ α + β From Fig. 5, for 1ABC ′
and 1ADC ′ we have γ ≤ 2π − (α + β). Eq. (10) satisfies bothinequalities simultaneously. Hence, proved. �
S.R. Sahoo et al. / Automatica 49 (2013) 519–525 525
Fig. A.1. A is the center of the sphere of radius AB. AB = AC = AD. BCD forms aspherical triangle on the sphere.
Appendix B. Proof of Lemma 3.5
At the stationary point, (θ ck + φk) =
ni=1(θ
ci +φi)
n . Now∇θ1,...,θn,β,λfmod =
sin(θ1 + φ1) − λ, . . . , sin(θn + φn) − λ,
−2λβ,−(θi + β2
− (n − 2)π). We now compute the second
derivative of fmod to characterize a maxima or minima.
∇2θ1,...,θn,β,λ
fmod|(θci +φi, ∀i)
=
cos(θ c1 + φ1) · · · 0 0 −1
.... . .
......
...0 · · · cos(θ cn + φn) 0 −10 · · · 0 −2λ 0
−1 · · · −1 0 0
.All the principal minors of −(∇2
θ1,...,θn,β,λfmod|(θci +φi, ∀i)) are posi-
tive. So −(∇2θ1,...,θn,β,λ
fmod|(θci +φi, ∀i)) is positive definite and hence,(∇2
θ1,...,θn,β,λfmod|(θci +φi, ∀i)) is a negative definite matrix. Thus, the
stationary point is a point of maxima. Now fmax = −n
i=1
cos(θ ci +φi) = −n cosn
i=1(θci +φi)
n
≤ −n cos
(n−2)π+
ni=1 φi
n
<
−n cos(n−2)π+nφ
n
. Hence, proved. �
References
Ando, H., Oasa, Y., Suzuki, I., & Yamashita, M. (1999). Distributed memorylesspoint convergence algorithm for mobile robots with limited visibility. IEEETransactions on Robotics and Automation, 15, 818–828.
Bernhart, A. (1959). Polygons of pursuit. Scripta Mathematica, 24, 24–50.Böhringer, K., Brown, R., Donald, B., Jennings, J., & Rus, D. (1997). Distributed
robotic manipulation: experiments in minimalism. In Oussama Khatib, &J.Salisbury (Eds.), Lecture notes in control and information sciences: vol. 223.Experimental robotics IV (pp. 11–25). Berlin, Heidelberg: Springer,http://dx.doi.org/10.1007/BFb0035193.
Brockett, R. W., & Liberzon, D. (2000). Quantized feedback stabilization of linearsystems. IEEE Transactions on Automatic Control, 45, 1279–1289.
Bruckstein, A. M., Cohen, N., & Efrat, A. (1991). Ants, crickets and frogs in cyclicpursuit. Technical report CIS report #9105. Computer Science Department.Israel Institute of Technology. July.
Delchamps, D. F. (1990). Stabilizing a linear system with quantized state feedback.IEEE Transactions on Automatic Control, 35, 916–924.
Elia, N., & Mitter, S. K. (2001). Stabilization of linear systems with limitedinformation. IEEE Transactions on Automatic Control, 46, 1384–1400.
Fredslund, J., & Matarić, M. J. (2002). A general algorithm for robot formations usinglocal sensing and minimal communication. IEEE Transactions on Robotics andAutomation, 18, 837–846.
Galloway, K. S., Justh, E. W., & Krishnaprasad, P. S. (2009). Geometry of cyclicpursuit. In Proceedings of the 48th IEEE conference on decision and control(pp. 7485–7490). Shanghai. December.
Galloway, K. S., Justh, E. W., & Krishnaprasad, P. S. (2010). Cyclic pursuit in threedimensions. In Proceedings of the 49th IEEE conference on decision and control(pp. 7141–7146). Atlanta. December.
Hui, Q. (2011). Finite-time rendezvous algorithms for mobile autonomous agents.IEEE Transactions on Automatic Control, 56, 207–211.
Ishii, H., & Francis, B. A. (2002). Stabilizing a linear system by switching control withdwell time. IEEE Transactions on Automatic Control, 47, 1962–1973.
Lin, J., Morse, A. S., & Anderson, B. D. O. (2007a). The multi-agent rendezvousproblem. Part 1: the synchronous case. SIAM Journal on Control and Optimization,46, 2096–2119.
Lin, J., Morse, A. S., & Anderson, B. D. O. (2007b). The multi-agent rendezvousproblem. Part 2: the asynchronous case. SIAM Journal on Control andOptimization, 46, 2120–2147.
Marshall, J. A., Broucke, M. E., & Francis, B. A. (2004a). Formation of vehicles in cyclicpursuit. IEEE Transactions on Automatic Control, 49(11), 1963–1974.
Marshall, J. A., Broucke, M. E., & Francis, B. A. (2004b). Unicycles in cyclic pursuit.In Proceedings of the 2004 American control conference, Vol. 6 (pp. 5344–5349).Boston, June.
Nair, G. N., & Evans, R. J. (2000). Communication-limited stabilization of linearsystems. In Proceedings of the 39th IEEE conference on decision and control, Vol. 1(pp. 1005–1010). Sydney, Australia, December.
Pavone, M., & Frazzoli, E. (2007). Decentralized policies for geometric patternformation and path coverage. Journal of Dynamic Systems, Measurement, andControl, 129, 633–643.
Petersen, I. R., & Savkin, A. V. (2001). Multi-rate stabilization of multivariablediscrete-time linear systems via a limited capacity communication channel.In Proceedings of the 39th IEEE conference on decision and control, Vol. 1(pp. 304–309). Orlando, USA. December.
Ramirez, J. L., Pavone, M., & Frazzoli, E. (2009). Cyclic pursuit for spacecraftformation control. In Proceedings of the 2009 American control conference(pp. 4811–4817). St Louis. June.
Ren, W., Beard, R. W., & Atkins, E. M. (2005). A survey of consensus problemsin multi-agent coordination. In Proceedings of the American control conference,Vol. 3 (pp. 1859–1864). Portland. June.
Richardson, T. (2001). Stable polygons of cyclic pursuit. Annals of Mathematics andArtificial Intelligence, 31, 147–172.
Sachs, S., LaValle, S. M., & Rajko, S. (2004). Visibility-based pursuit-evasion in anunknown planar environment. The International Journal of Robotics Research, 23,3–26.
Sinha, A., & Ghose, D. (2006). Generalization of linear cyclic pursuit withapplications to rendezvous of multiple autonomous agents. IEEE Transactionson Automatic Control, 51(11), 1819–1824.
Tovar, B., Guilamo, L., & LaValle, S. M. (2005). Gap navigation trees: minimalrepresentation for visibility-based tasks. In Michael Erdmann, Mark Overmars,David Hsu, & Frank van der Stappen (Eds.), Springer tracts in advanced robotics:vol. 17. Algorithmic foundations of robotics VI (pp. 425–440). Berlin, Heidelberg:Springer, http://dx.doi.org/10.1007/10991541_29.
Tovar, B., LaValle, S. M., &Murrieta, R. (2003). Optimal navigation and object findingwithout geometric maps or localization. In Proceedings of IEEE internationalconference on robotics and automation, Vol. 1 (pp. 464–470).Mexico City,Mexico,September.
Tovar, B., Murrieta-Cid, R., & LaValle, S. M. (2007). Distance-optimal navigation in anunknown environmentwithout sensing distances. IEEE Transactions on Robotics,23, 506–518.
Yu, J., LaValle, S. M., & Liberzon, D. (2008). Rendezvous without coordinates.In Proceedings of the IEEE conference on decision and control (pp. 1803–1808).Cancun, December.
Soumya Ranjan Sahoo received his B.Tech degree fromthe University College of Engineering, Burla, India in Elec-trical Engineering in 2008. He is currently a ResearchScholar in Systems and Control Engineering, IIT Bombay,India. His research interests includemulti-agent coordina-tion and control.
Ravi N. Banavar received his B.Tech. in Mechanical Engi-neering from IIT Madras (1986), his Masters (Mechanical,1988) and Ph.D. (Aerospace, 1992) degrees from ClemsonUniversity and the University of Texas at Austin, respec-tively. He had a brief teaching stint at UCLA in 1991–92,soon after which he joined the Systems and Control Engi-neering group at IIT Bombay in early 1993. He is currentlya professor in this group and has spent a few sabbaticalbreaks during the years at UCLA (Los Angeles), IISc (Banga-lore) and LSS (Supelec, France.) His research interests arebroadly in the field of geometric mechanics and nonlinear
control, with applications in electromechanical and aerospace engineering prob-lems.
Arpita Sinha received the Ph.D. degree from the Depart-ment of Aerospace Engineering, Indian Institute of Science,Bangalore, India in 2007. In 2008, she was a postdoctoralresearcher at Cranfield University, Shrivenham, UK. Since2009, she has been anAssistant Professorwith the Systemsand Control Engineering, Indian Institute of Technology,Bombay, India. Her research interests include multivehi-cle coordination and control, guidance strategies for au-tonomous vehicles and task allocation in multi-agent sys-tems.
