Robotics and Autonomous Systems 60 (2012) 1149–1164
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Robotics and Autonomous Systems
journal homepage: www.elsevier.com/locate/robot
Pattern preserving path following of unicycle teams with communication delays
Qin Li a,∗, Zhong-Ping Jiang b
a Statoil Research Center, Porsgrunn 3936, Norwayb Department of Electrical and Computer Engineering, Polytechnic Institute of New York University, Brooklyn, NY 11201, USA
a r t i c l e i n f o
Article history:Received 27 March 2009Received in revised form2 March 2012Accepted 15 May 2012Available online 31 May 2012
Keywords:Multi-vehicle controlConnectivityPotential functionNonlinear control
a b s t r a c t
This paper examines the problem of pattern-preserving path following control for unicycle teams withtime-varying communication delay. A key strategy used here introduces a virtual vehicle formationsuch that each real vehicle has a corresponding virtual vehicle as its pursuit target. Under an input-driven consensus protocol, the virtual vehicle formation is forced to stay close to the desired vehicleformation; and a novel controller design is proposed to achieve virtual leader tracking for each vehiclewith constrained motion. It is shown that, by the proposed strategy, the pattern can be preserved if theformation speed is less than some computable value that decreases with increasing size of delay, and theexact desired formation pattern can be eventually achieved if this speed tends to zero.
Formation control design for multi-vehicle systems continuesto attract great attention due to the needs in many industrialand military applications such as surveillance, search, rescue andterrain mapping. Trajectory tracking and path following withformation maintenance are two problems under wide study. Intrajectory tracking (resp. path following), one vehicle or somegeometric characteristics of a vehicle team is required to tracka virtual vehicle moving on the given trajectory (resp. to followa given path). With the formation maintenance requirement,the geometric pattern of a vehicle team needs to be (globally)asymptotically stabilized at a desired one, which either is givenby the relative positions among the vehicles, or maps to a value(e.g. global or local minimum) of some given function (e.g. artificialpotential function). For real-world applications, theremay be someextra control objectives which need to be achieved. For instance,inter-vehicle and vehicle–obstacle collisions should be avoided inthe transient of the tracking or path following.
Several types of strategies have been proposed for the controlpurposes stated above during the past few years. The authors of[1–9] investigated leader–follower structure based strategies,where the vehicle group is layered and each vehicle in some layerhas a vehicle in the upper layer as the local leader to follow; and the
This work was supported in part by the NSF grants ECS-0093176 and DMS-0504462, and in part by the NNSF of China under grant 60628302.∗ Corresponding author.
only vehicle in the top layer is required to track a given trajectoryor follow a given path when the group is performing formationtracking or a path following task. Artificial potential function (APF)based approaches were firstly employed for the swarming andflocking control of multiple vehicles with holonomic dynamics[10–14]. These approaches have recently proved useful also fornonholonomic vehicle teams [15–19]. By the APF based strategy,each vehicle in a team tries to follow the direction specified bythe negative gradient of corresponding APF component, and thegeometric pattern of the team almost converges to the one thatmaps to a local minimum of the collective APF. For holonomicvehicles, this following can be exactly realized at any time; but itmay only be achieved asymptotically for nonholonomic vehicles.The main difficulty of the APF based method is to design an APFwithout local minima which correspond to undesirable patterns.Another important method for formation control of multi-vehiclesystems is based on the use of the so called virtual structure,which is composed of virtual leaders playing the role of referencetargets for the real vehicles. These virtual leaders can be in rigidconfigurations, interactwith each other for some formation controlpurposes, or interconnect their motions with those of the realvehicles. Early work along this line can be found in [20,21]. Recentyears have witnessed much effort in applying this strategy totrajectory tracking and path following of multiple vehicles withvarious types of dynamics [22–25,19,26,27]. See [28] for morereferences on the subject of formation control.
Although many previous works have studied the formationcontrol of nonholonomic vehicles, very few of them dealt withdata transmission delay in inter-vehicle communication channels.Among the latter, [25] showed a decentralized strategy in which
1150 Q. Li, Z.-P. Jiang / Robotics and Autonomous Systems 60 (2012) 1149–1164
each vehicle follows a path traced out by a virtual leader,and the desired pattern of the vehicles can be eventuallyachieved if the individual coordination states reach consensus.In [29], the authors designed distributed control laws basedon backstepping techniques such that a team of vehicles areforced to asymptotically form a desired pattern, with respectto a global coordinate system, whose centroid moves along adesired trajectory. An appealing feature of the approach is that thedesired trajectory only needs to be available to a portion of thevehicles. Note that in both the papers the communication delay isconsidered time-invariant, and the topology of the communicationgraph of the vehicle team is not related to its geometric pattern.
In this paper, we address, for the first time, the problem of howto drive a nonholonomic vehicle team to move along some givenpath with a preserved pattern and a specified formation speedprofile in the presence of time-varying communication delay. Thepattern of the vehicle team is said to be preserved if, roughlyspeaking, the error between it and the desired one remains ina small range. Considering distance-dependent communicationcapability of the vehicles, we assume that the communicationnetwork of the vehicle team is fixed, bidirectional and connected ifthe pattern is preserved. Our main contribution is to give a controlstrategy by which the pattern is preserved if the desired formationspeed is upper bounded, where the bound is shown inverselyproportional to the size of delay. The strategy can deal with boththe cases that the desired formation speed is bounded away fromzero and converges to zero asymptotically.
Applying a virtual structure framework, we assign each vehiclewith a path reference point (PRP) on the given path, based onwhich a virtual lead is defined. The formation of the virtual leaders,called the virtual formation, coincides with the desired vehicleformation if all the PRPs reach agreement. The proposed strategycan be outlined as follows. On one hand, an input-driven consensusprotocol is employed to keep PRPs having small differences andproceeding roughly with the desired formation speed. In otherwords, the resulting virtual formation remains close to the desiredvehicle formation. On the other hand, under the action of acontrol law derived from the APF based approach, each vehicleasymptotically tracks its virtual leader with the tracking errorconstrained inside a pre-defined range. As a consequence, thereal vehicle formation is always close to the virtual formation.The combination of the above two points guarantees that theactual vehicle formation has a small error with respect to thedesired one. Last but not least, by our approach, when the desiredformation speed converges zero (e.g., in the scenario of point-to-point migration) the team of vehicles eventually form exactly thedesired formation.
2. Preliminaries
Throughout this paper, we use N, R+, Z+ to denote the setsof natural numbers, nonnegative real numbers and nonnegativeintegers. ∥x∥ denotes the Euclidean norm of the vector x ∈ Rn
for any n ∈ N. In addition, we use 0E to represent the functionmapping any point in an interval E ⊆ R to 0 ∈ RN , where thedimension N ∈ N can be identified from the context.
2.1. Graph theory
A directed graph G(V, E) consists of a vertex set V and an arc,or directed edge, set E ⊂ V × V . For any i, j ∈ V , the ordered pair(i, j) ∈ E if and only if i is a neighbor of j. Vertex i is said to havea self edge if (i, i) ∈ E . A directed path, with length n − 1, fromvertex i to j is a sequence of distinct vertices v1, v2, . . . , vn, wheren ≥ 1, v1 = i, vn = j and (v1, v2), . . . , (vn−1, vn) ∈ E . A directedgraph is said to have a spanning tree if and only if there exists a
Fig. 1. An example of artificial potential functions.
vertex i ∈ V , called the root, such that there is a directed path fromi to any other vertex. A graph G(V, E) is undirected if and only iffor any i, j ∈ V , (i, j) ∈ E implies (j, i) ∈ E . A path in an undirectedgraph is defined analogously as a directed path in a directed graph.An undirected graph is said to be connected if and only if there is apath between anypair of vertices. The degree di of vertex i ∈ V in anundirected graph G(V, E) is defined as di = Card(j : (i, j) ∈ E);and the value of maxi di is called themaximum degree of the graph.The diameter of a connected undirected graphG(V, E) is defined tobe maxi,j∈V Lij, where Lij is the minimum length of any path fromvertex i to vertex j. In addition, for the graph G(V, E) with time-dependent edge set E(t), we use
t G(V, E(t)) to represent the
graph composed of node set V and edge set
t E(t). See [30] formore basics in graph theory.
2.2. Artificial potential function
An artificial potential function V (·, r) : [0, r) → [a, ∞), withr > 0 and a ∈ R+, used in this paper has the following properties:
(a) V ∈ C2;(b) limx→r V (x, r) = ∞;(c) ∀ϵ ∈ (0, r), ∃ δ > 0 such that V ′(x, r) ≥ δ, ∀x ∈ [ϵ, r).
In the following, we use V ′x and V ′′
x to denote the first and secondderivatives of the function V (x, r) with respect to x. An example ofartificial potential functions is depicted in Fig. 1.
3. Pattern preserving path following
3.1. Problem statement and description of the control strategy
The path to be followed by the vehicle team is a plane curverepresented by the function q : R → R2. We use xq(s), yq(s)to denote the first and second components of q(s) respectively,i.e., q(s) = (xq(s), yq(s)). Physically, xq(s), yq(s) are the x- andy-coordinates of the point corresponding to s on the path withrespect to some right-handed Cartesian coordinate system Σg .Regarding the smoothness of the path, we make the followingAssumption 1:
Assumption 1. The derivatives x′q(s), y′
q(s), x′′q(s), y′′
q(s), x′′′q (s),
y′′′q (s) exist and are bounded, and there exists a positive real
constant c such that
(x′q(s))2 + (y′
q(s))2 ≥ c , for any s ∈ R.
An example of such a path is xq(s) = 10 cos(0.1s) and yq(s) =
10 sin(0.1s). Assumption 1 will be mainly used to ensure someproperties of themotion of the virtual leaders that the real vehiclesare supposed to track.
Q. Li, Z.-P. Jiang / Robotics and Autonomous Systems 60 (2012) 1149–1164 1151
Fig. 2. A desired formation of the vehicle team.
Let θ(s) be the tangential angle of the path at q(s) and hence
cos(θ(s)) =x′q(s)
(x′q(s))2 + (y′
q(s))2,
sin(θ(s)) =y′q(s)
(x′q(s))2 + (y′
q(s))2,
(1)
θ ′(s) =y′′q(s)x
′q(s) − x′′
q(s)y′q(s)
(x′q(s))2 + (y′
q(s))2. (2)
For each point with the position q(s) on the path, define alocal right-handed Cartesian coordinate system Σs whose origincoincides with q(s), x-axis aligns with the unit tangent vector[cos(θ(s)), sin(θ(s))]⊤, and z-axis has the same directionwith thatof the global coordinate system Σg (see Fig. 2).
The vehicle team we consider consists of N unicycles withkinematic model:
xi = vi cosφi,
yi = vi sinφi, i = 1, 2, . . . ,N, (3)φi = ωi,
where pi := (xi, yi) is the position of vehicle i in the globalframe Σg , φi is the vehicle’s heading with respect to the x-axisof Σg , vi is the vehicle’s velocity along its body axis, and ωi is itsangular velocity (See Fig. 5, wherein the vehicle index is omitted).In the rest of the paper, the symbol V is used to denote the set1, 2, . . . ,N.
The vehicle team is regarded to follow the path qwith a desiredpattern li : i ∈ V, li : R → R2, and a speed profile vl : [t0, ∞) →
R, called the formation speed, if the vehicles maintain the desiredoffset li with respect to a common reference point on the pathmoving with the speed vl, or, phrased in more formal words, ifthere exists a time-dependent function s : [t0, ∞) → R, satisfyings(t) = vl(t) (with some initial value at the initial time t0), such thatpi(t) = pri(s(t)), with
pri(s) := q(s) + Φ(s)li(s), (4)
for any i ∈ V and any t ≥ t0, whereΦ is the rotationmatrix definedas
Φ(s) =
cos(θ(s)) − sin(θ(s))sin(θ(s)) cos(θ(s))
.
In (4), q(s) is the position of the reference point on the path, andthe desire offsets li(s) are expressed in the local coordinate systemΣs (see Fig. 2). The vehicle team is said to asymptotically follow
Fig. 3. Illustration of the formation path following control strategy.
the path with the desired pattern if limt→∞(pi(t) − pri(s(t))) = 0and limt→∞(s(t) − vl(t)) = 0. It is assumed here that the offsetsli are continuous. Note that we allow the desired offsets to varyalong the path, which may be useful for the purpose of inter-vehicle or vehicle–obstacle collision avoidance (see also [23] forsome remarks on this point).
Now, associate each vehicle i ∈ V with an individual pathreference point (PRP) on the given path whose position is q(si) =
(xq(si), yq(si)), where si ∈ R is called the path parameter of vehiclei. For our control purposes, the path parameter si varies with timeand can be real-timeupdated by vehicle i (for example, by using theconsensus algorithm studied in Section 3.2). Each vehicle i has li(si)as its desired offset with respect to its PRP, expressed in the localcoordinate Σsi . Here, we suppose that each vehicle i has perfectknowledge about the path and its desired offset with respect to itsown PRP; i.e., it knows the exact values of q(si), Φ(si) and li(si). Inthis case, each vehicle i can set up a virtual leader which is amovingpoint with the position pri(si), defined in (4), and heading φri(si)given by
cos(φri(si)) =x′
ri(si)(x′
ri(si))2 + (y′
ri(si))2,
sin(φri(si)) =y′
ri(si)(x′
ri(si))2 + (y′
ri(si))2,
(5)
where xri(si) and yri(si) denote the first and second elements ofpri(si) respectively (see Fig. 3 for an illustration). Later on, inSection 3.4, we shall impose some restrictions on the desiredoffsets li such that x′
ri(si) and y′
ri(si) cannot be both zero for any si.Note that given i ∈ V , for each j ∈ V ,
It follows that limt→∞ pj(t) − prj(si(t)) = 0 if limt→∞(si(t) −
sj(t)) = 0, and limt→∞(pj(t) − prj(sj(t))) = 0. Therefore,the vehicle team asymptotically follow the path with the desiredpattern li and formation speed vl, with the function s(t) = si(t), ifwe further have limt→∞(si(t) − vl(t)) = 0. When communicationdelay exists, the asymptotic consensus of the path parameters,i.e., limt→∞(si(t) − sj(t)) = 0, may not be expected. But weknow from the above analysis that pj(t) − prj(si(t)) is arbitrarilyclose to zero, or, roughly speaking, the pattern of the vehicleteam is arbitrarily close to the desired one if |si(t) − sj(t)| and∥pj(t) − prj(t)∥ are both small enough for any j ∈ V , or, inother words, if the PRPs remain close enough to each other and
1152 Q. Li, Z.-P. Jiang / Robotics and Autonomous Systems 60 (2012) 1149–1164
a
b
Fig. 4. (a) Four vehicles achieving a PPFPF with l1 = (a, a), l2 = (a, −a), l3 =
(−a, a), l1 = (−a, −a). (b) Three vehicles achieving an AFPF with l1 = (b, 0),l2 = (−b/
√2, b/
√2), l3 = (−b/
√2, −b/
√2). In both (a) and (b), ‘‘v.l.’’ is short
for ‘‘virtual leader’’, the arrows on the paths indicate the directions along which theargument of the path function q increases, the desired speed vl is positive, a and bare some positive reals, and the times t0 , t1 and t2 satisfy t0 < t1 < t2 .
each vehicle stays close enough to its virtual leader (see Fig. 3).This observation motivates us to propose the following pattern-preserving formation path following and asymptotic formationpath following problems:Pattern-preserving formation path following (PPFPF) and asymptoticformation path following (AFPF): Given a desired formation speedprofile vl, a set of desired offsets li : i ∈ V, and positiveconstants ϵk, k ∈ 1, 2, 3, 4with ϵ1 < ϵ2, the PPFPFwith the triple(li, vl, ϵk) is said to be achieved if whenever, for each i, j ∈ V ,
the AFPF with the triple is said to achieved if, furthermore,
limt→∞
(si(t) − sj(t)) = 0, (12)
limt→∞
(si(t) − vl(t)) = 0. (13)
The vehicle team is said to preserve the pattern li across[t0, t∗], with t∗ ≥ t0, if and only if, for all t ∈ [t0, t∗], (8) holdsfor every i and j and (10) holds for every i.
Fig. 4(a) and 4(b) illustrate two vehicle teams achieving a PPFPFand an AFPF respectively. Note that the PRPs of vehicles do notnecessarily converge for the PPFPF.
Remark 1. Note that (11) and (12) are the main control objectivesin [23], but therein communication delay is not considered.
We will achieve (8) and (9) by using the so called averagingconsensus protocols (see Section 3.2), and construct the controlsvi and ωi, i ∈ V , for (10) and (11) based on the artificial potentialfunctions (see Section 3.3). For the first objective, the values of siwill be transmitted from vehicle to vehicle. We consider here thatthe vehicles have limited communication capacities, so that some
vehicle may not receive the data from another which is far away. Itis be assumed, however, that the communication graph (definedat the beginning of the next subsection) of the vehicle team isfixed, undirected and connected when the pattern of the vehicleteam is preserved (see Section 3.4). This assumption is reasonableif the communication graph is undirected and connected when thevehicle team is with the desired pattern and the establishment ofa communication link between two vehicles depends on the inter-vehicle distance.
3.2. Continuous-time averaging consensus systems with time delaysand external input
The aggregation of the PRPs, as expressed in (8) and (9), isrealized by the following averaging consensus protocol: For anyi ∈ V and t ∈ [t0, ∞),
si(t) =
Nj=1
aij(t)sj(t − τ i
j (t)) − si(t)+ vl(t), (14)
where aij(t) are called the weighting factors; τ ij (t) represents the
transmission delay of the data received by vehicle i at time t ,which was sent from vehicle j; and the desired formation speedvl(t) is considered here as the external input. Due to limitedcommunication capacity, at time t , each vehicle i can receive datafrom vehicles in the set Ni(t) ⊂ V , called Ni(t) the neighbor set ofvehicle i (as reflected in Assumptions 3 and 4). For any i ∈ V , welet i ∈ Ni(t) and assume that τ i
i (t) = 0, which means that eachvehicle i knows the current value of si. In addition, it is supposedthat no data is generated before the initial time t0, which impliesthat the functions τ i
j (t) satisfy t−τ ij (t) ≥ t0. Note that a simplified
protocol was used in [25], where the communication delay wasconsidered as a constant. See [31–36] for more discussions on theconsensus problems.
Let G(V, E(t)), called the communication graph, be the graphcomposed of the vertex set V and the time-dependent edge setE(t) := (j, i) : i ∈ V, j ∈ Ni(t). In the following, wewill first givea result about the protocol (14) in a general setting (Theorem 1),where the neighbor set Ni(t) for each i ∈ V can vary over time.While the scenario with a fixed and connected graph G(V, E(t)),which will be addressed immediately after (Theorem 2), is morerelevant to our discussion of the PPFPF and AFPF in this paper, thestudy on the general problemdeserves a place not only because it isof independent interest but also because it builds up a foundationfor investigating the special case.
We start with some assumptions on the protocol (14).
Assumption 2 (Dwell Time). There exists an infinite sequence oftimes t0, t1, . . ., with tk+1 > tk, such that the edge set E(t) isinvariant in [tk, tk+1) for all k ∈ Z+, where the length of all the timeintervals [tk, tk+1), k ∈ Z+, are uniformly lower bounded abovezero, i.e.,
infk∈Z+
tk+1 − tk > ∆dw > 0.
We will use T to denote the set tk : k ∈ Z+. Assumption 2implies that the communication graph G(V, E(t)) changes onlyfinite times in any finite time period.
Assumption 3. For each i, j ∈ V , the weighting factor aij(t)satisfies
aij(t) ∈ [a⋆, a⋆], if j ∈ Ni(t),
aij(t) = 0, otherwise,
where a⋆ and a⋆ are two positive real numbers, and is continuousover each time interval on which j ∈ Ni(t).The weighting factors indicate how a vehicle weighs the informa-tion received from its neighbors in the update of its path parame-ter.
Q. Li, Z.-P. Jiang / Robotics and Autonomous Systems 60 (2012) 1149–1164 1153
Assumption 4. There exists some positive real number Bd suchthat, for any i, j ∈ V , the communication delay τ i
j (t) satisfies
τ ij (t) ∈ [0, Bd], if j ∈ Ni(t),
τ ij (t) = 0, otherwise;
and is continuous over each time interval on which j ∈ Ni(t).
Assumption 4 says that if vehicle iutilizes data fromvehicle j (i.e. sj)at time t then the data must be sent from vehicle j no earlier thant − τ i
j (t). The value of Bd is called the size of the communicationdelay.
Assumption 5 (Connectivity). There exists Tc ∈ R+ such that forany t ∈ [t0, ∞), the graph
Assumptions 5 and 6 are imposed to ensure sufficient informationexchange for the consensus. Assumption 5 is a standard connec-tivity condition used in solving consensus problems, see, e.g., [37]and [34]. Assumption 6 says that if, at some point in time, vehi-cle j uses data from vehicle i for the update of sj then, conversely,vehicle i should be able to use the data from vehicle j for the up-date of si within a time period of certain length. In other words, theinter-vehicle information exchange is reciprocal.
Assumption 7. vl(t) is continuous over [t0, ∞).
Remark 2. The (piecewise) continuity assumption on aij(t), vl(t)and τ i
j (t) guarantees the existence and uniqueness of a solution of(14) over any interval [t0, tf ], tf ≥ t0, with arbitrarily given initialcondition at t0, where by such a solution we mean a function thatsatisfies (14) with right-sided derivative (see Section 8 in [38]).
In the rest of the paper, for notational simplicity, define the timeinterval
Et := [maxt0, t − Bd, t], ∀t ≥ t0. (15)
In particular, Et0 = t0. Now, let us first state a general result aboutthe protocol (14) under Assumptions 2–7.
Theorem 1. Suppose Assumptions 2–7 hold. Then by protocol (14),∀t ∈ [T + KT ⋆, T + (K + 1)T ⋆), ∀T ∈ [t0, ∞), and ∀K ∈ Z+,
Ω(t) ≤ γ KΩ(T ) + b · supτ∈ET∪[T ,t]
|vl(τ )|, (16)
where
Ω(t) := maxi∈V
maxτ∈Et
si(τ ) − mini∈V
minτ∈Et
si(τ ), t ∈ [t0, ∞), (17)
b = 2Bd
1 + a⋆T ⋆ 2 − γ
1 − γ
, (18)
and
a⋆= Na⋆, (19)
T ⋆= 2NB + (N − 1)Tc, (20)
B = maxBr , Bd + ∆dw, (21)
γ = 1 −
a⋆
a⋆
N−1(1 − e−a⋆∆dw )N−1e−a⋆(2N−1)B. (22)
Proof. See Appendix.
We may interpret Ω(t) as the consensus error of the pathparameters (or the aggregation error of the PRPs). Also note that
Ω(t0) = maxi∈V si(t0) − mini∈V si(t0). Thus, Theorem 1 givesan explicit upper bound for the consensus error which containsone part (exponentially decaying) related with its initial value andanother with the input vl, both affected by the communicationdelay. It can be seen that the consensus error is ultimatelydominated by the latter. If the input vl tends to zero as timegoes to infinity, then the the path parameters reach consensusasymptotically. That is, we have the following result:
Corollary 1. Suppose Assumptions 2–7 hold. Then by protocol (14),limt→∞ Ω(t) = 0 and, in particular, limt→∞ |si(t) − sj(t)| = 0, iflimt→∞ vl(t) = 0.
Corollary 2 below is a technical result that will be used inSection 3 for the realization of the AFPF.
Corollary 2. Suppose Assumptions 2–7 hold, and vl(t) is differen-tiable on [t0, ∞). Then by protocol (14),
∞
t0|si(t)|dt < ∞, for all
i ∈ V if
∞
t0|vl(t)| + |vl(t)|dt < ∞
The proofs of the two corollaries are put in Appendix.The following Theorem 2 enables us to impose upper bounds
on the desired formation speed vl and the weighting factors aijsuch that the pattern preserving requirements (8) and (9) in thePPFPF are met, with initial condition (6), if the communicationgraph is fixed, undirected, and connected. Furthermore, the boundscan be computed based on the diameter and the maximum degreeof the graph and the size of the communication delay. Note thatAssumptions 2, 5 and 6 hold if the communication graph has thisproperty.
Theorem 2. Suppose Assumptions 3, 4 and 7 hold, and the graphG(V, E(t)) is fixed, undirected, and connected for all t ∈ [t0, ∞),with r1, d1 the diameter and the maximum degree respectively. Forany positive real number L1, if
a⋆≤
L1d1Bd
, (23)
then, for any t ∈ [T + KT ⋆, T + (K + 1)T ⋆), T ≥ t0, and K ∈ Z+,
Ω(t) ≤ γ KΩ(T ) + L2 supτ∈ET∪[T ,t]
|vl(τ )|, (24)
where Ω has been defined in (17),
T ⋆=
2r1L1d1a⋆
, (25)
γ = 1 −
a⋆
d1a⋆
r1−1
(1 − e−L1)r1−1e−(2r1−1)L1 , (26)
L2 = 2Bd
1 +
2L1r1(2 − γ )
1 − γ
. (27)
In particular, if Ω(t0) ≤ ϵ1 and, for any t ≥ t0,
|vl(t)| <ϵ2 − ϵ1
L2, (28)
where ϵ1 and ϵ2 are two positive real numbers such that ϵ2 > ϵ1, thenΩ(t) < ϵ2 for all t ≥ t0. If, furthermore, for any given ϵ3 > 0,
a⋆≤
ϵ3
d1ϵ2, (29)
then |si(t) − vl(t)| ≤ ϵ3 for all t ≥ t0.
Proof. See Appendix.
Note that, in Theorem 2, γ ∈ (0, 1) and L2 > 0.It can be seen that, given the size of the communication delay
Bd, (28) imposes an upper bound on the desired formation speed vl,
1154 Q. Li, Z.-P. Jiang / Robotics and Autonomous Systems 60 (2012) 1149–1164
in terms of the characteristics of a fixed, undirected and connectedcommunication graph, so that (8) can be achieved under the initialconstraint (6); the smaller the value of L2 the larger the formationspeed is allowed. Clearly, the value of L2 depends on the boundson the weighting factors, i.e. a⋆ and a⋆, and the freely chosenquantity L1. It is easy to see that a⋆ = a⋆ gives the smallest γ ,hence the least L2, for any selection of L1. This implies that positiveweighting factors in (14) should be set equal; and the left freedomfor minimizing L2 is to tune L1. On the latter point, the minimizer,denoted by L⋆
1, can be explicitly solved from (26) and (27). Whatshould be emphasized is that L2 cannot be made arbitrarily smallsince it approaches ∞ as L1 tends either to 0 or ∞ (as γ → 1). Inaddition to the constraint on vl, one sees from (23) and (29) thatthe weighting factors need to be smaller than min
L1d1Bd
,ϵ3
d1ϵ2 to
realize both (8) and (9).The property of the communication graph in Theorem 2 will
be guaranteed by the combination of the consensus algorithm andvirtual leader tracking controller introduced next; see Section 3.4for a detailed explanation.
3.3. Virtual leader tracking controller
In this section, we propose the controller for each vehicle totrack its virtual leader with constrained motion, i.e. to achievethe goals in (10) and (11). For this purpose, we first establish aunicycle model for the motion of the virtual leader of each vehicle;then with the application of the backstepping designmethodology(see, e.g., [39]) a feedback controller will be synthesized to driveeach vehicle to asymptotically track its the virtual leader (i.e., (11)is satisfied). Furthermore, by incorporating artificial potentialfunction in the design, the tracking features the novel propertythat themotion of each vehicle is restricted inside a specified circlecentered at the position of its virtual leader, so that the controlobjective (10) is realized.
As mentioned in Section 3.1 the virtual leader of vehicle i hasthe position pri(si) = (xri(si), yri(si)) and the heading angle φri(si),given in (4) and (5). Note from (5) that if
(x′
ri(si))2 + (y′
ri(si))2 = 0(this will be guaranteed by Lemma 1 in Section 3.4) the dynamicsof (xri(si), yri(si)) and φri(si) can be described by
xri = vri cosφri,
yri = vri sinφri, (30)φri = ωri,
where
vri =
(x′
ri)2 + (y′
ri)2 · si,
ωri =y′′
rix′
ri − x′′
riy′
ri
(y′
ri)2 + (x′
ri)2
· si.(31)
The model (30)–(31) is the unicycle model of the virtual leader ofvehicle i.
To facilitate our control design, it is supposed here that thevelocities vri and ωri satisfy the following assumption (The vehicleindex i is omitted):
Assumption 8. The velocities vr and ωr are continuous andbounded, and either of the following two conditions holds:(C1) vr does not change its sign for all t ≥ t0, and vr,min ≤ |vr | ≤
vr,max, where vr,max ≥ vr,min are positive real constants;(C2)
Remark 3. Assumption 8will be ensured by the constraints on thedesired formation speed and vehicle offsets, which are specified inthe hypothesis H1–H3 and Lemma 1 in Section 3.4 respectively;see the proof of Theorem 4 for a detailed justification.
Fig. 5. Tracking error expression in the body frame of a vehicle: The (right-handed)body frame of the vehicle Σb has its x-axis along the vehicle’s body axis and z-axisparallel to that of the global frame Σg . The thick arrow represents the vector withthe expression (x − xr , y − yr ) in Σg .
Inspired by [40] and [41], we now design vi and ωi for eachvehicle i using the backstepping methodology to achieve (10) and(11) under the initial condition (7). Since the design is identical foreach vehicle, we drop the vehicle index in the subscript of eachvariable that is concerned (e.g., xr instead of xri will be used).
To start the control design, define the tracking error asxeyeφe
=
cos(φ) sin(φ) 0− sin(φ) cos(φ) 0
0 0 1
x − xry − yrφ − φr
.
Note that xe, ye, and φe are the position and heading tracking errorsexpressed in the body frame of a vehicle, as illustrated in Fig. 5.Clearly, (11) is ensured if xe, ye, and φe vanishes as time goes toinfinity; (10) holds if x2e + y2e ≤ ϵ2
4 .By (3) and (30), we know that the errors xe, ye, and φe have the
defined in Section 2.2, and ϵ4 a positive real constant. Without lossof generality, the minimum value of the function V is consideredto be zero.
Applying the backstepping techniques, we first give a desiredvalue for the heading difference φe:
αφe = − arcsin
g(t)V ′ye
1 + (V ′xe)
2 + (V ′ye)
2
, (32)
where
V ′
xe :=∂V (ρ(xe, ye), ϵ4)
∂xe= V ′
ρxe,
V ′
ye :=∂V (ρ(xe, ye), ϵ4)
∂ye= V ′
ρye.
The function g will be chosen following the idea that (i) |g(t)| < 1for all t ≥ t0 so that the definition in (32) is always valid, and (ii)the errors xe and ye would be driven asymptotically to zero by anappropriate linear velocity v when we have φe = αφe . Then, by
Q. Li, Z.-P. Jiang / Robotics and Autonomous Systems 60 (2012) 1149–1164 1155
backstepping, the angular velocity will be cooked to guarantee thediminishing of φe − αφe .
Let ze = φe − αφe , we rewrite the dynamics of xe and ye as
xe = v − vr cos(ze + αφe) + ωye= v + ωye + fx, (33)
where
h1 =
1 + (V ′
xe)2 + (V ′
ye)2,
h2 =
1 + (V ′
xe)2 + (1 − g2(t))(V ′
ye)2,
fx = −vrcos(ze)h−1
1 h2 + sin(ze)h−11 g(t)V ′
ye
;
and
ye = vr sin(ze + αφe) − ωxe= −vrh−1
1 g(t)V ′
ye − ωxe + fy,
where
fy = vr(1 − cos(ze))h−11 g(t)V ′
ye + vr sin(ze)h−11 h2.
By choosing
v = −k1V ′
xe − fx, (34)
where the constant k1 > 0, it follows that V = −k1(V ′xe)
2−
vrh−11 g(t)(V ′
ye)2 if φe = αφe . It can be shown that V is bounded
(hence x2e + y2e < ϵ24 ) and limt→∞ xe = limt→∞ ye = 0 if the
function g is given as
g(t) = h−11 (k2sgn(vr) + k3 sin(k4t)) , (35)
with the constants k2, k3, k4 > 0 satisfying k2 + k3 < 1 andk2vr,min > k3vr,max, if the condition C1 in Assumption 8 holds; or
g(t) = h−11 (k5 sin(k6t)) , (36)
with the constants k5 ∈ [0, 1) and k6 = 0, if the condition C2 holds.Now, following the idea of backstepping, we design the angular
velocity ω so that, in addition to xe and ye, the error between φeand its desired value αφe converges to zero asymptotically, whichimplies limt→∞ φe = 0 (as limt→∞ αφe = 0). Note that the errorze = φe − αφe has the dynamics:
ze = φe − αφe
= ω − ωr + h−12
g(t)V ′
ye + g(t)V ′
ye
− h−1
2 h−11 h1g(t)V ′
ye
=1 − h−1
2 g(t)V ′
xe
ω + fz,
where
fz = −ωr + h−12
g(t)V ′
ye + g(t)V ′′
ρ ρye
+ V ′
ρ(−vrh−11 g(t)V ′
ye + fy)
− h−12 h−1
1 h1g(t)V ′
ye .
Consider the function
W = V
ρ(xe, ye),12ϵ24
+
12z2e . (37)
With the linear velocity v given in (34), we have
W = −k1(V ′
xe)2− vrh−1
1 g(t)(V ′
ye)2
+ ze
1 − h−1
2 g(t)V ′
xe
ω + fz
+ V ′
yevrh−11
1 − cos(ze)
zeg(t)V ′
ye +sin(ze)
zeh2
.
Thus, setting
ω =1
1 − h−12 g(t)V ′
xe
−k7ze − fz − V ′
yevrh−11
×
1 − cos(ze)
zeg(t)V ′
ye +sin(ze)
zeh2
, (38)
with the constant k7 > 0, leads to
W = −k1(V ′
xe)2− vrh−1
1 g(t)(V ′
ye)2− k7z2e . (39)
This property of the function W plays a key role in proving thefollowing main result on the virtual leader tracking:
Theorem 3. Suppose Assumption 8 holds, and the initial distancebetween each vehicle and its virtual leader is strictly less than ϵ4,i.e., x2e (t0) + y2e (t0) < ϵ2
4 . Then x2e (t) + y2e (t) < ϵ24 for any
t ∈ [t0, ∞), and limt→∞ xe(t) = limt→∞ ye(t) = limt→∞ φe = 0if the velocities v andw are given as in (34) and (38) respectively, withany k1, k7 > 0 and the function g chosen as in (35) or (36) dependingon which condition in Assumption 8 is satisfied.
Proof. See Appendix.
3.4. Pattern preserving path following
In this section, the two designs presented in Sections 3.2 and3.3 are integrated to give solutions to the problems of PPFPF andAFPF with the triple (li : i ∈ V, vl, ϵk : k = 1, 2, 3, 4) whileputting some restrictions on the desired formation speed vl and thedesired pattern li. As mentioned in Section 3.1, a key assumptionwe make is that the communication graph of the vehicle teamG(V, E(t)) is fixed, undirected and connected if the pattern of theteam is preserved, i.e., if (8) and (10) are satisfied for each i, j ∈ V(the last assumption in Theorem 4). This actually implies that thegraph G(V, E(t0)) is undirected and connected if initially (6) and(7) hold, and G(V, E(t)) = G(V, E(t0)) for all t ≥ t0 if the PPFPF isachieved.
Before going into the details of the main result, let us firstroughly explain how the two designs in Sections 3.2 and 3.3work jointly to realize the PPFPF. First, the results in Section 3.2(Theorem 2, Corollaries 1 and 2) allow us to specify constraints onthe desired formation speed vl, in terms of the characteristics of theinitial communication graph and the size of the communicationdelay, under which the control goals (8) and (9) can be achievedwith the initial condition (6) and the Assumption 8 for the virtualleader tracking can be satisfied. Second, the result on the virtualleader tracking (Theorem 3) guarantees the control objectives(10) and (11) be met with the initial condition (7). On theother hand, the realization of (8) and (10) in turn ensures thatthe communication graph of the vehicle team be always fixed,undirected, and connected.
According to Theorem 2we first impose an upper bound on thedesired formation speed vl(t). The rationale behind this has beenstated at the end of Section 3.2 (after Theorem 2).
H1: |vl(t)| ≤ vl,max <ϵ2−ϵ1
L2, and vl(t) is bounded over [t0, ∞),
where L2 is computed from (27) by choosing some L1, with r1 and d1being the diameter andmaximumdegree of the graphG(V, E(t0)).
Under H1, we discuss the following two scenarios:H2: |vl(t)| ≥ vl,min > 0 for all t ∈ [t0, ∞).H3:
∞
t0|vl(t)| + |vl(t)|dt < ∞, and vl(t) is bounded for all
[t0, ∞).The following technical lemma specifies the restrictions on the
desired vehicle offsets li that, together with the above restrictionson vl, will be used to fulfill the conditions on the motion of virtualleaders in Assumption 8.
1156 Q. Li, Z.-P. Jiang / Robotics and Autonomous Systems 60 (2012) 1149–1164
Lemma 1. Suppose Assumption 1 holds. Then there exist c2i, c3i > 0such that, for any si ∈ R,
c2i ≤
(x′
ri(si))2 + (y′
ri(si))2 ≤ c3i,
and x′′
ri(si), y′′
ri(si) are bounded if either of the following two conditionsholds:
(i) lxi(si) and lyi(si) are constants, and there exist c4i, c5i > 0 suchthat either |lxi(si)| ≥ c4i or |lyi(si) − κ(si)| ≥ c5i for all si ∈ R;
(ii) lxi(si) is constant, lyi(si), l′yi(si), l′′
yi(si) are bounded, and there existc6i, c7i > 0 such that either |lyi(si) − κ(si)| ≥ c6i or |l′yi(si) +
lxi(si)θ ′(si)| > c7i, for all si ∈ R;
where li(si) = (lxi(si), lyi(si)) is the desired offset of vehicle i at si, andκ(si) = sgn(θ ′(si))σ (si), with θ ′(si) given in (2) and σ(si) the radiusof curvature of the the given path at the point q(si).
where c2i, c3i are as in Lemma 1, and ϵ3 = minϵ3, vl,min − c5,with c5 an arbitrary real constant in (0, vl,min). Note that vri,max >vri,min ≥ c2ic5.
Now we are in a position to present our main result on thepattern preserving path following of unicycle teams.
Theorem 4. Suppose all the following assumptions hold:
(1) Assumption 1 holds for the given path to be followed by the vehicleteam.
(2) For the desired offset li, either of the two conditions inLemma 1 holds.
(3) The communication delay τ ij (t) satisfies Assumption 4.
(4) The weighting factors aij(t) satisfy Assumption 3, with a⋆≤
min
L1d1Bd
,ϵ3
d1ϵ2
, where d1 is the maximum degree of the graph
G(V, E(t0)), Bd is the size of the communication delay (as inAssumption 4), L1 is as in H1, and ϵ3 is as in (40).
(5) The graph G(V, E(t)) is fixed, undirected, and connected over[t0, t], for any t ≥ t0, if the vehicle team preserves its patternacross [t0, t]; i.e., if, for any t ∈ [t0, t], (8) holds for every i, j ∈ Vand (10) for each i ∈ V .
Then, under the action of the consensus protocol (14) and the trackingcontroller (34) and (38), the PPFPF with the triple (li : i ∈
V, vl, ϵk : k = 1, 2, 3, 4) is achieved if H1 and H2 hold and thefunction gi(t) is set as in (35); the AFPF is achieved if H1 and H3 holdand gi(t) is set as in (36).
Proof. In this proof, the notations xei, yei, φei, vri, ωri, k5i, andWi, when omitting the vehicle index ‘‘i’’, have been defined inSection 3.3; they all correspond specifically to vehicle i here.
Suppose H1 and H2 hold.We first show thatG(V, E(t)) is fixed,undirected, and connected over [t0, ∞). By contradiction, if thiswas not the case, then by the assumption (5), either (8) or (10)would be violated in finite time; hence
Besides, by the continuity of pi(t)−pri(t) and si(t), we have, for allt ∈ [t0, t∗], maxi,j |si(t) − sj(t)| ≤ ϵ2 and maxi ∥pi(t) − pri(t)∥ ≤
ϵ4. It follows that G(V, E(t)) is fixed, undirected, and connectedacross t ∈ [t0, t∗]. By (6), Ω(t0) = maxi,j |si(t0) − sj(t0)| ≤ ϵ1,where Ω is defined in (17). It follows from Theorem 2 that Ω(t) <ϵ2 and therefore maxi,j |si(t) − sj(t)| < ϵ2 for all t ∈ [t0, t∗].On the other hand, from the definition of t∗, we know that either
maxi,j |si(t∗) − sj(t∗)| = ϵ2 or there exists some j ∈ V such that∥pj(t∗)−prj(t∗)∥ = ∥(xej(t∗), yej(t∗))∥ = ϵ4 (or both). Thus, it mustbe the case that ∥(xej(t∗), yej(t∗))∥ = ϵ4 for some j ∈ V . We nowshow that this is in fact not possible. From Theorem 2, we knowthat |sj(t) − vl(t)| ≤ ϵ3 for all t ∈ [t0, t∗]. Consequently, by (31),(40), H1, H2 and Lemma 1, it follows that vrj, ωrj are bounded, and
vrj,min ≤ |vrj| ≤ vrj,max,
for all t ∈ [t0, t∗]. Therefore, by Theorem 3, we have that∥(xej(t∗), yej(t∗))∥ < ϵ4.
As we now know that G(V, E(t)) is fixed, undirected, andconnected over [t0, ∞), it follows from Theorems 2 and 3 that(8)–(11) hold for all t ∈ [t0, ∞) and each i, j ∈ V , i.e., the PPFPF isachieved.
Next, assume that H1 and H3 hold. Similarly as above, wefirst show, by contradiction, that G(V, E(t)) = G(V, E(t0)) forall t ≥ t0. Define t∗ as in (41). Then, again, we only need todisprove that ∥(xej(t∗), yej(t∗))∥ = ϵ4 for some j ∈ V . It iseasy to see from Corollary 2 that, for any i ∈ V ,
t∗t0
|si(t)|dt <
∞. Then, by (31), (39), and Lemma 1, we have that, for eachi ∈ V , Wi(t) ≤ k5ic3i|si(t)| < ∞, which implies Wi(t∗) ≤
Wi(t0) + t∗t0
k5ic3i|si(t)|dt < ∞. By the property (b) of theartificial potential function V (used in the definition ofWi in (37)),∥(xei(t∗), yei(t∗))∥ < ϵ4.
Since G(V, E(t)) = G(V, E(t0)) for all t ∈ [t0, ∞), fromTheorem 2, we have that (8) and (9) are realized. Note that H1 andH3 imply limt→∞ vl = 0 and limt→∞ vl = 0. By Corollary 1, thisgives limt→∞ Ω(t) = 0 andhence limt→∞ maxi,j |si(t)−sj(t)| = 0.Furthermore, according to the protocol (14), we have limt→∞ si =
limt→∞ vl = 0. On the other hand, it follows from (31) andLemma 1 that vri and ωri are both bounded and limt→∞ vri(t) =
limt→∞ ωri(t) = 0. In addition, by Corollary 2 and Lemma 1,∞
t0|vri(t)| < ∞. Consequently, by virtue of Theorem 3, we have,
for any i ∈ V , ∥(xei(t), yei(t))∥ < ϵ4, for all t ∈ [t0, ∞), andlimt→∞ xei(t) = limt→∞ yei(t) = limt→∞ φei(t) = 0. In summary,the AFPF is achieved.
It can be seen that the conditions (1)–(4) in Theorem 4actually set conditions on the path geometry, desired offsets,communication delay, and the weighting factors respectively. Thecondition on the path geometry can be satisfied by a path planning;the one on the desired offsets restricts the choice of the desiredpattern; and the bounds on theweighting factors cause no problemsince they are free to choose. As to communication delay, accordingto Theorem 4, if the size of communication delay Bd is given, thento achieve the PPFPF or AFPF the desired formation speed needs tobe upper bounded as in H1, where the maximally allowed upperbound can be obtained by finding the value of L1 that minimizesL2 using (27) and (26). On the other hand, if a range of desiredformation speed is given, then it has to be ensured that L2 can bemade small enough. Referring to (27), this may demand limitationon the size of communication delay,which concerns the design andimplementation of the communication systemaboard the vehicles.
4. Simulation results
In this section, we verify our path following strategy bycomputer simulations. We first consider a scenario where fourunicycles are required to follow a sinusoidal path with xq(s) = s,yq(s) = 10 sin(0.1s). The desired formation pattern is specified bythe desired offsets l1 = (1, 1), l2 = (1, −1), l3 = (−2, 2), l4 =
(−2, −2). The specifications in the PPFPF problem are set as ϵ1 =
0.4, ϵ2 = 2, ϵ3 = 0.04 and ϵ4 = 3. By Theorem 2, if thebound for communication delays is given by Bd = 1ms, thenthe weighting factors aij(t) should be upper bounded by 0.01, and
Q. Li, Z.-P. Jiang / Robotics and Autonomous Systems 60 (2012) 1149–1164 1157
Fig. 6. Motion snapshots of a 4-vehicle team following a sinusoidal path.
the desired formation speed |vl(t)| needs to be less than vl,max =
0.2m/s. However, to show the conservativeness of our results, welet vl(t) = 0.4 sin(0.2t) + 0.6, and the communication delays arechosen randomly in the discrete set 0.1, 0.2, . . . , 1.0 (unit: s) ateach updating time. (In our program, the updating step is set as0.1s.) In addition, the controller for the virtual leader tracking isdesigned using the approach introduced in Section 3.3 (see (34)and (38)), where the potential function is cooked as
V (x, r) =1
x + 1+
r2
r − x,
g(t) is as in (35), and ki, i = 1, 2, 3, 4, 7 are chosen as 2, 1.5, 0.04,1 and 2 respectively.
The simulation results are shown in Figs. 6–9: Fig. 6 shows somesnapshots of themotion of the vehicle team; Figs. 7 and 8 illustratethe evolutions of maxi,j∈V |si(t) − sj(t)| and maxi∈V |si(t) − vl(t)|respectively; while Fig. 9 depicts the convergence of the virtualleader tracking errors. It is clearly seen from these figures that therequirements (8)–(11) are all satisfied. The periodic behavior of thespeed error as seen in Fig. 8 can be briefly explained as follows. Foreach i = 1, 2, 3, 4, si(t) remains very close to vl(t) (theoretically,the error is bounded by ϵ3 = 0.04.), hence si(t) is basically theintegration of vl(t), which is the sum of a linear function plus asinusoidal function. Now it should be clear that the sinusoidal partof |si(t) − vl(t)| = |
j∈Ni
aij(sj(t − τ ij (t)) − si(t))| is due to the
existence of the delays.Next, in Figs. 10 and 11, we present the simulation results
with another two scenarios. Fig. 10 shows the motion snapshotsof a 4-vehicle team following a circle with radius 10 m (pathfunction: xq(s) = 10 cos(0.1s), yq(s) = 10 sin(0.1s)). The desiredpattern is a square with the offsets l1 = (1, 1), l2 = (1, −1),l3 = (−1, 1), l4 = (−1, −1). Initial conditions are set as
Fig. 11 depicts the motion snapshots of a 3-vehicle team followingthe same sinusoidal path as in the first simulation. But here thedesired pattern is triangular with the offsets l1 = (2, 0), l2 =
(0, −2), l3 = (0, 2). Initial conditions are put as
1158 Q. Li, Z.-P. Jiang / Robotics and Autonomous Systems 60 (2012) 1149–1164
m
axi
||
pi (
t) -
pri
(si (
t))||
21 2
Fig. 9. Value of maxi 12 ∥pi(t) − pri(si(t))∥2 .
Fig. 10. Motion snapshots of a 4-vehicle team following a circle.
In both the simulations, the desired formation speed vl, ϵi’s, andother simulation parameters are set the same as in the previousone.
5. Concluding remarks
In this paper we have proposed a cooperative strategy for thepattern preserving path following of a unicycle team with time-varying communication delays. Our key assumption is that thecommunication graph of the vehicle team is fixed, undirected, andconnected when the pattern is preserved. The main result of thispaper indicates that to prevent the vehicle formation from havinglarge distortion, its movement along the path must slow downas the communication delay becomes larger. Indeed, an explicitrelation is given between the desired formation speed and the sizeof communication delay.
Another assumption we make in the present paper is that eachvehicle has perfect knowledge about the path and thedesired offsetwith respect to its path reference point. This might not be practicalfor applicationswith large-scale vehicle teams. One solution to thisissue is a two-level control strategy: On the first level, the vehiclesthat are informed of the path and offsets information can use thecontrol strategy proposed in this paper to achieve path followingwith a desired pattern, which serves as the skeleton pattern forthe overall pattern of the whole team. On the second level, theother vehicles are grouped and each of the groups forms somespecified pattern with respect to one of the 1st-level vehicles. Forthe pattern preserving on the second level, either leader-followingor leaderless cooperative control strategies may be adopted, yetdetailed control laws are still open for design if time-varyingcommunication delay has to be taken into account (the resultsin [29] may be used when the delay is constant).
Interesting topics for future investigation also include howto design a distributed control scheme to meet additionalcontrol objectives, such as obstacle avoidance and point-to-pointmigration with timing constraints, and how to achieve the PPFPFor AFPF with constraints on control effort.
Appendix. Proofs
Proof of Theorem 1. We first study the dynamics of si(t) :=
si(t) − tt0
vl(τ )dτ , which can be written as
˙si(t) =
Nj=1
aij(t)sj(t − τ i
j (t)) − si(t)+ ξi(t),
∀i ∈ V, ∀t ≥ t0, (42)
where
ξi(t) = −
Nj=1
aij(t) t
t−τ ij (t)
vl(τ )dτ . (43)
Fig. 11. Motion snapshots of a 3-vehicle team following a sinusoidal path.
Q. Li, Z.-P. Jiang / Robotics and Autonomous Systems 60 (2012) 1149–1164 1159
Clearly, the function ξi is continuous on [tk, tk+1), ∀k ∈ Z+.We use s(t; ET , ϕ) = [s1(t; ET , ϕ), . . . , sN(t; ET , ϕ)]⊤, T ≥ t0,
to denote a solution of (42) on some interval [T , Tf ], Tf ≥ T , withcontinuous initial function ϕ = [ϕ1, . . . , ϕN ], ϕi : ET → R, whereET is defined in (15), i.e., s(t; ET , ϕ) is continuous, equal to ϕ(t) forall t ∈ ET , and satisfies (42) on [T , Tf ] (right-sided derivative beingtaken). As commented in Remark 2, under Assumptions 2–4 and 7,such a solution exists and is unique for any T ≥ t0 and any Tf ≥ T .
Define the functions Mi(t; ET , ϕ), mi(t; ET , ϕ), M(t; ET , ϕ) andm(t; ET , ϕ), t ≥ T , as follows:
Mi(t; ET , ϕ) = maxτ∈Et
si(τ ; ET , ϕ),
mi(t; ET , ϕ) = minτ∈Et
si(τ ; ET , ϕ),(44)
M(t; ET , ϕ) = maxi∈V
Mi(t; ET , ϕ),
m(t; ET , ϕ) = mini∈V
mi(t; ET , ϕ).(45)
Note that since Et0 = t0, we have Mi(t0; Et0 , ϕ) = mi(t0;Et0 , ϕ) = ϕi, M(t0; Et0 , ϕ) = maxi∈V ϕi, and m(t0; Et0 , ϕ) =
mini∈V ϕi.The proof of Theorem 1 is based on the following Lemmas 2
and 3.Roughly speaking, Lemma 2 shows that the difference between
si and sj, for any i and j, over time never exceeds the integral of acommon bound on the magnitudes of the input functions ξi.
Lemma 2. Suppose that Assumptions 2–4 and 7 hold. Let ζ (·) be anon-negative continuous function satisfying
|ξi(t)| ≤ ζ (t), ∀t ≥ t0, ∀i ∈ V. (46)
Then, for any T ∈ [t0, ∞) and any t ≥ T ,
M(t; ET , 0ET ) − m(t; ET , 0ET ) ≤ 2 t
Tζ (τ )dτ , (47)
Proof. Define ηi(t) = si(t; ET , 0ET ) − tT ζ (τ ) + ϵdτ , and let ϵ be
any positive real number. We show that ηi(t) ≤ 0 for all i ∈ Vand all t ≥ T . By contradiction, suppose this was not true and lett⋆ = inft ∈ [T , ∞) : ∃ i ∈ V s.t. ηi(t) > 0. Then, we musthave some i ∈ V with ηi(t⋆) = 0 and ηi(t⋆) ≥ 0 (the derivative istaken right-sided). We see that at time t⋆ there is a non-empty setW ⊂ V such that ηi(t⋆) = 0 for all i ∈ W and ηi(t⋆) < 0 for alli ∈ V \W . Indeed, if ηi(t⋆) < 0, ∀i ∈ V , then by continuity of ηi(t),there is some t1 > t⋆ such that ηi(t) < 0, ∀t ∈ [t⋆, t1], ∀i ∈ V ,which contradicts with the definition of t⋆; while if ηi(t⋆) > 0 forsome i ∈ V , then there must be some t2 < t⋆ such that ηi(t) > 0for all t ∈ [t2, t⋆], which also contradicts the definition of t⋆. Nowif ηi(t⋆) < 0, ∀i ∈ W , then it is easy to conclude that there is somet3 > t⋆ such that ηi(t) < 0 for all t ∈ (t⋆, t3] and all i ∈ V , whichis again a contradiction.
Without loss of generality, let η1(t⋆) = 0, η1(t⋆) ≥ 0. Note thatfor any t ∈ [T , t⋆] and any i ∈ V \ 1, ηi(t) ≤ η1(t⋆), i.e.,
si(t; ET , 0ET ) −
t
Tζ (τ ) + ϵdτ
≤ s1(t⋆; ET , 0ET ) −
t⋆
Tζ (τ ) + ϵdτ , ∀t ∈ [T , t⋆],
which gives si(t; ET , 0ET ) ≤ s1(t⋆; ET , 0ET ) for all i ∈ Vand t ∈ [T , t⋆]. And consequently, from (42), we know that˙s1(t⋆; ET , 0ET ) ≤ ξ1(t⋆) < ζ(t⋆) + ϵ, i.e., η1(t⋆) < 0, acontradiction.
Since ϵ can be an arbitrarily positive constant, we conclude thatsi(t; ET , 0ET ) ≤
tT ζ (τ )dτ for any i ∈ V and any t ∈ [T , ∞).
On the other hand, by a similar reasoning, it can be proved thatsi(t; ET , 0ET ) ≥ −
tT ζ (τ )dτ for all i ∈ V and all t ∈ [T , ∞). And
(47) follows from the definitions in (44) and (45).
Next, we study the homogeneous system of (42):
˙sui (t) =
j∈V\i
aij(t)suj (t − τ i
j (t)) − sui (t). (48)
As for the nonhomogeneous system (42), we use su(t; ET , ϕ) =
[su1(t; ET , ϕ), . . . , suN(t; ET , ϕ)]⊤, T ≥ t0, to denote the (unique)solution of (48) with continuous initial function ϕ = [ϕ1, . . . , ϕN ],ϕi : ET → R. In addition, define the functions Mu
i (t; ET , ϕ),mu
i (t; ET , ϕ), Mu(t; ET , ϕ) andmu(t; ET , ϕ), t ≥ T , as follows:
Mui (t; ET , ϕ) = max
τ∈Etsui (τ ; ET , ϕ),
mui (t; ET , ϕ) = min
τ∈Etsui (τ ; ET , ϕ),
Mu(t; ET , ϕ) = maxi∈V
Mui (t; ET , ϕ),
mu(t; ET , ϕ) = mini∈V
mui (t; ET , ϕ).
The following Lemma 3 basically says that the maximaldifference among sui over time is exponentially decaying andalways bounded by the maximal difference among their initialfunctions.
Lemma 3. Suppose Assumptions 2–7 hold. Then we have for any T ∈
[t0, ∞),
Mu(t; ET , ϕ) − mu(t; ET , ϕ)
≤ maxi∈V
maxτ∈ET
ϕi(τ ) − mini∈V
minτ∈ET
ϕi(τ ), ∀t ∈ [T , T + T ⋆), (49)
Mu(T + T ⋆; ET , ϕ) − mu(T + T ⋆
; ET , ϕ)
≤ γ
maxi∈V
maxτ∈ET
ϕi(τ ) − mini∈V
minτ∈ET
ϕi(τ )
, (50)
where T ⋆ and γ are defined in (20) and (22) respectively.
Proof. To ease the notations, denote sui (t; ET , ϕ),Mu(t; ET , ϕ) andmu(t; ET , ϕ) simply by sui (t), M
u(t) and mu(t) respectively. Since
Mu(T ) = maxi∈V
maxτ∈ET
ϕi(τ ), mu(T ) = mini∈V
minτ∈ET
ϕi(τ ),
in order to show (49), it suffices to justify that Mu(t) is non-increasing, andmu(t) is non-decreasing for all t ≥ T . Let T2 > T1 ≥
T , and for t ∈ [T1, T2], define µi(t) = sui (t) − Mu(T1) − tT1
ϵdτ ,where ϵ is any positive real constant. Using the same reasoningas in the proof of Lemma 2, it can be shown that µi(t) ≤ 0 forall t ∈ [T1, T2]. Since ϵ can be arbitrarily small, we have thatsui (t) ≤ Mu(T1) for all t ∈ [T1, T2], and hence Mu(T2) ≤ Mu(T1).The monotonicity ofmu(t) can be proved in a similar way.
Next, we show the inequality (50). It is clear that there existssome i1 ∈ V and some t ′0 ∈ [T , T + Bd] such that sui1(t
′
0) =
Mu(T + Bd). Let tm = mintk ∈ T : tk > t ′0, where T is defined inAssumption 2. Since the neighbor set Ni1(t) is constant in the timeinterval [t ′0, tm), for all t ∈ [t ′0, tm) we have,
˙sui1(t) =
j∈Ni1 (t ′0)
ai1j(t)(suj (t − τ
i1j (t)) − sui1(t))
≥
j∈Ni1 (t ′0)
ai1j(t)(mu(t) − sui1(t))
≥ a⋆(mu(t ′0) − sui1(t)), (51)
1160 Q. Li, Z.-P. Jiang / Robotics and Autonomous Systems 60 (2012) 1149–1164
where a⋆ is given in (19) and we have employed the monotonicityof the function mu. As a result, by the Comparison Lemma ([39],Lemma 3.4), for all t ∈ [t ′0, tm),
sui1(t) ≥ σ + e−a⋆(t−t ′0)ρ,
where
σ := mu(t ′0), ρ := Mu(T + Bd) − mu(t ′0). (52)
Following the same procedure, this inequality can be extended fort ≥ tm, i.e., for all t ∈ [t ′0, ∞),
sui1(t) ≥ σ + e−a⋆(t−t ′0)ρ. (53)
Noting that B > Bd, where B is defined in (21), it is easy to obtainfrom (53) that
mui1(t
′
0 + B) ≥ σ + e−a⋆Bρ. (54)
Now, let Q1 ⊂ V be the set of agents such that i ∈ Q1 ifand only if mu
i (t′
0 + B) ≥ σ + e−a⋆Bρ. (Obviously, i1 ∈ Q1.) Lettk1 = mintk ∈ T : tk > t ′0 + B. Here we need to deal with twocases: (a) tk1 − (t ′0 + B) > ∆dw; and (b) tk1 − (t ′0 + B) ≤ ∆dw .For Case (a), by Assumption 5, there must be some time instant t ∈
[t ′0+B, t ′0+B+Tc] and some agentsw1 ∈ Q1,w2 ∈ V\Q1 such thateither (w2, w1) ∈ E(t) or (w1, w2) ∈ E(t) (or maybe both hold).Then theremust be some time t1 ∈ t ′0+B∪(T ∩[t ′0+B, t ′0+B+Tc])such that, over [t1, t1 + ∆dw], E(t) remains invariant and either(w2, w1) ∈ E(t) or (w1, w2) ∈ E(t). Further, let t1 be the smallestin these times. Note that there is no connection between the agentsin Q1 and Q2 over [t ′0 + B, t1) (or, in other words, they can be seenas two isolated groups in this period). Consequently, in the sameway how we prove the monotonicity of mu, it can be shown thatmini∈Q1 m
ui (t) is non-decreasing over [t ′0 + B, t1) and hence
mui (t1) ≥ σ + e−a⋆Bρ (55)
for all i ∈ Q1.Now, suppose firstly (w2, w1) ∈ E(t1). Following a similar
derivation as in (51) and again noting the monotonicity of mu, wehave for all i ∈ Q1 and all t ∈ [t1, ∞),
˙sui (t) ≥ a⋆(σ − sui (t)), (56)
which gives that for all i ∈ Q1 and all t ∈ [t1, t1 + ∆dw],
sui (t) ≥ σ + e−a⋆(B+∆dw)ρ.
This, together with (55), shows that mui (t) ≥ σ + e−a⋆(B+∆dw)ρ
for any i ∈ Q1 and any t ∈ [t1, t1 + ∆dw]. As a result, for allt ∈ [t1, t1 + ∆dw],
˙suw2
(t) =
j∈Nw2 (t1)
aw2j(t)(suj (t − τ
w2j (t)) − suw2
(t))
≥ aw2w1(suw1
(t − τw2w1
(t)) − suw2(t))
+
j∈Nw2 (t1)\w1
aw2j(t)(σ − suw2(t))
= aw2w1(suw1
(t − τw2w1
(t)) − σ)
+
j∈Nw2 (t1)
aw2j(t)(σ − suw2(t))
≥ a⋆e−a⋆(B+∆dw)ρ + a⋆(σ − suw2(t)), (57)
which, combining with suw2(t1) ≥ σ , implies
suw2(t1 + ∆dw) ≥ σ +
a⋆
a⋆(1 − e−a⋆∆dw )e−a⋆(B+∆dw)ρ.
Again, a similar analysis as in (51) gives that ˙suw2
(t) ≥ a⋆(σ −
suw2(t)) for all t ∈ [t1 + ∆dw, t1 + 2B], which follows that for any
t ∈ [t1 + ∆dw, t1 + 2B],
suw2(t) ≥ σ +
a⋆
a⋆(1 − e−a⋆∆dw )e−a⋆(t−t1+B)ρ.
Therefore,
muw2
(t1 + 2B) ≥ σ +a⋆
a⋆(1 − e−a⋆∆dw )e−a⋆3Bρ. (58)
On the other hand, we obtain from (55) and (56) that for all i ∈ Q1,
mui (t1 + 2B) ≥ σ + e−a⋆3Bρ. (59)
(58) and (59) together show that for any i ∈ Q1 ∪ w2,
mui (t1 + 2B) ≥ σ +
a⋆
a⋆(1 − e−a⋆∆dw )e−a⋆3Bρ. (60)
Now, suppose (w1, w2) ∈ E(t1), then according to Assump-tion 6, there exists a time interval [t1, t1 + ∆dw] ∈ [t1, t1 + B]such that (w2, w1) ∈ E(t) for any t ∈ [t1, t1 +∆dw]. From (56) and(55), we have for all i ∈ Q1 and all t ∈ [t1, t1 + ∆dw],
mui (t) ≥ σ + e−a⋆(t1+∆dw−t1+B)ρ.
Then, by the same reasoning as in (57), it can be concluded that forall t ∈ [t1, t1 + ∆dw],
˙suw2
(t) ≥ a⋆e−a⋆(t1+∆dw−t1+B)ρ + a⋆(σ − suw2(t)),
which follows
suw2(t1 + ∆dw) ≥ σ +
a⋆
a⋆(1 − e−a⋆∆dw )e−a⋆(t1+∆dw−t1+B)ρ.
In consequence,we can easily obtain that for any t ∈ [t1+∆dw, t1+2B],
suw2(t) ≥ σ +
a⋆
a⋆(1 − e−a⋆∆dw )e−a⋆(t−t1+B)ρ.
Therefore we can still reach (58) and (60).Next, we discuss Case (b). By Assumption 5, theremust be some
time t ′1 ∈ T ∩ [tk1 , tk1 + Tc] and some agents w1 ∈ Q1, w2 ∈
V \ Q1 such that, over [t ′1, t′
1 + ∆dw], either (w2, w1) ∈ E(t) or(w1, w2) ∈ E(t) (or maybe both hold). We further let t ′1 be thesmallest in these times. Repeating some procedures before, it is notdifficult to derive that mu
i (t′
1) ≥ σ + e−a⋆(tk1−t ′0)ρ for all i ∈ Q1.Then, by the same reasoning as in the proof for Case (a), we canachieve that for any i ∈ Q1 ∪ w2,
mui (t
′
1 + 3B − (tk1 − t ′0)) ≥ σ +a⋆
a⋆(1 − e−a⋆∆dw )e−a⋆3Bρ. (61)
Now, note that in Case (a),
t1 + 2B ≤ (t ′0 + B + Tc) + 2B = t ′0 + 3B + Tc;
while in Case (b),
t ′1 + 3B − (tk1 − t ′0) ≤ (tk1 + Tc) + 3B − (tk1 − t ′0)= t ′0 + 3B + Tc .
Hence (60) and (61) show that there exists some time t1 ∈ [t ′0, t′
0+
3B + Tc] such that for all i ∈ Q1 ∪ w2,
mui (t1) ≥ σ +
a⋆
a⋆(1 − e−a⋆∆dw )e−a⋆3Bρ.
Using similar arguments after (54), we conclude that there is atime t ∈ [t ′0, t
′
0 + 2(N − 1)B + (N − 1)Tc] such that,
mu(t) ≥ σ +
a⋆
a⋆
N−1(1 − e−a⋆∆dw )N−1e−a⋆(2N−1)Bρ
= σ + (1 − γ )ρ,
Q. Li, Z.-P. Jiang / Robotics and Autonomous Systems 60 (2012) 1149–1164 1161
where γ is defined in (22). Lastly, it follows from themonotonicityofMu(t) and mu(t) that
Mu(T + T ⋆) − mu(T + T ⋆) ≤ Mu(T + Bd) − mu(t) ≤ γ ρ
≤ γ (Mu(T ) − mu(T )),
where T ⋆ is defined in (20).
Using Lemmas 2 and 3, we now prove Theorem 1.Let TK = T + KT ⋆, K ∈ N. By the linearity of system (42), we
have for all T ≥ t0 and all t ∈ [T , T1),
s(t; ET , ϕ) = su(t; ET , ϕ) + s(t; ET , 0ET ), (62)
1162 Q. Li, Z.-P. Jiang / Robotics and Autonomous Systems 60 (2012) 1149–1164
Combining (70)–(72), it is easy to show that ∀t ∈ [T + KT ⋆, T +
(K + 1)T ⋆), ∀T ∈ [t0, ∞), and ∀K ∈ Z+,
Ω(t) ≤ γ KΩ(T ) + 2Bd
1 + a⋆T ⋆ 2 − γ
1 − γ
· sup
τ∈ET∪[T ,t]|vl(τ )|.
Proof of Corollary 1. Firstly, since vl(t) is continuous over [t0, ∞)and convergent as t goes to infinity, we know that vl(t) is boundedover [t0, ∞). Then, by setting T = t0 in (16), it follows that Ω(t) isbounded over [t0, ∞). Then, again by (16), for any ϵ > 0, we canfind large enough T and K such that Ω(t) < ϵ for t ≥ T + KT ⋆.This completes the proof.
Proof of Corollary 2. By (14), we see that
|si(t)| ≤ a⋆Ω(t) + |vl(t)|, t ∈ [t0, ∞),
where Ω(t) has been defined in (17). As a result, to prove∞
t0|si(t)|dt < ∞, we only need to show
∞
t0Ω(t)dt < ∞.
By repeating the use of (16) at each T = t0 + KT ⋆, K ∈ Z+, itcan be shown that for all t ∈ [t0 + KT ⋆, t0 + (K + 1)T ⋆), K ∈ Z+,
Ω(t) ≤ γ KΩ(t0) + b
K−1l=0
γ K−1−l supτ∈Et0+lT⋆∪Il
|vl(τ )|
+ supτ∈Et0+KT⋆∪IK
|vl(τ )|
= γ KΩ(t0) + bΓK + bΓK ,
where
IK := [t0 + KT ⋆, t0 + (K + 1)T ⋆], K ∈ Z+,
ΓK =
K−1l=0
γ K−1−l supτ∈Et0+lT⋆∪Il
|vl(τ )|,
ΓK = supτ∈Et0+KT⋆∪IK
|vl(τ )|. (73)
Clearly, it is sufficient to show that
∞
K=0 ΓK + ΓK < ∞. But notethat for anym ∈ N, we have
mK=0
ΓK =
m−1l=0
supτ∈Et0+lT⋆∪Il
|vl(τ )| ·
mK=l+1
γ K−1−l
<1
1 − γ
m−1l=0
supτ∈Et0+lT⋆∪Il
|vl(τ )|. (74)
Further, it follows from the fact that Et0+KT⋆ ⊂ IK−1 for all K ∈ Nthat
supτ∈Et0+KT⋆∪IK
|vl(τ )| ≤ supτ∈IK−1
|vl(τ )| + supτ∈IK
|vl(τ )|, ∀K ∈ N. (75)
By (73)–(75), we see that the claim
∞
K=0 supτ∈IK |vl(τ )| < ∞willconclude the proof.
For any m ∈ Z+,m
K=0
IKsupτ∈IK
|vl(τ )|dτ −
mK=0
IK
|vl(τ )|dτ
≤
mK=0
IKsupτ∈IK
|vl(τ )| − infτ∈IK
|vl(τ )|dτ . (76)
By the continuity of vl(t), we know that there exist τK1, τK2 ∈ IKsuch that supτ∈IK |vl(τ )| = |vl(τK1)| and infτ∈IK |vl(τ )| = |vl(τK2)|.Therefore, using (76), we obtain that for allm ∈ Z+,m
K=0
IKsupτ∈IK
|vl(τ )|dτ −
mK=0
IK
|vl(τ )|dτ
≤
mK=0
IK
|vl(τK1)| − |vl(τK2)|dτ
≤
mK=0
T ⋆|vl(τK1) − vl(τK2)|
≤
mK=0
T ⋆
τK2
τK1
vl(τ )dτ ≤ T ⋆
mK=0
IK
|vl(τ )| dτ
≤ T ⋆
∞
t0|vl(τ )|dτ < ∞.
The claim is justified by further noting thatIKsupτ∈IK
|vl(τ )|dτ = T ⋆ supτ∈IK
|vl(τ )|dτ ,
mK=0
IK
|vl(τ )|dτ <
∞
t0|vl(τ )|dτ < ∞.
Proof of Theorem 2. Firstly, one notes that Lemma 2 still holds.Next, we prove a variation of Lemma 3 under the assumption ofthe theorem, that is:
Mu(t; ET , ϕ) − mu(t; ET , ϕ)
≤ maxi∈V
maxτ∈ET
ϕi(τ ) − mini∈V
minτ∈ET
ϕi(τ ), ∀t ∈ [T , T + T ⋆), (77)
Mu(T + T ⋆; ET , ϕ) − mu(T + T ⋆
; ET , ϕ)
≤ γ
maxi∈V
maxτ∈ET
ϕi(τ ) − mini∈V
minτ∈ET
ϕi(τ )
, (78)
where T ⋆ and γ are given in (25) and (26); and the other notationsare inherited from the lemma. Let Mu(t) and mu(t) be short forMu(t; ET , ϕ) and mu(t; ET , ϕ). Indeed, the first inequality followsfrom the properties that the functions Mu and mu are non-increasing and non-decreasing over [T , ∞) respectively, as shownat the beginning of the proof of Lemma 3. Now, let i ∈ V be suchan agent that, at some t ′0 ∈ [T , T + Bd], sui (t
′
0) = Mu(T + Bd). Then,again like in the proof of Lemma 3, one can show that
mui1(t
′
0 + B) ≥ σ + e−L1ρ,
where B =L1
d1a⋆≥ Bd, and σ and ρ are given in (52). Suppose
that Q ⊂ V be the set of agents such that i ∈ Q if and only ifmu
i (t′
0 + B) ≥ σ + e−L1ρ. (Obviously, i1 ∈ Q1.) Since the graphGpro is fixed, undirected, and connected, theremust be some agentsw1 ∈ Q, w2 ∈ V \ Q such that (w2, w1) ∈ E(t ′0 + B). It followsthat, for any i ∈ Q ∪ w2,
mui (t
′
0 + 3B) ≥ σ +a⋆
d1a⋆(1 − e−L1)e−3L1ρ.
Repeating the procedure, we arrive at
mui (t
′
0 + (2r1 − 1)B) ≥ σ +
a⋆
d1a⋆
r1−1
× (1 − e−L1)r1−1e−(2r1−1)L1ρ.
Again, noting the monotonicity ofMu(t) and mu(t), we have
Mu(T + T ⋆) − mu(T + T ⋆) ≤ Mu(T + Bd) − mu(t ′0 + (2r1 − 1)Bd)
≤ γ ρ
≤ γ (Mu(T ) − mu(T )),
which is exactly (78). The rest of the proof for (24) follows the samearguments as in that for Theorem 1 and is hence omitted.
Q. Li, Z.-P. Jiang / Robotics and Autonomous Systems 60 (2012) 1149–1164 1163
Now, if |vl(t)| <ϵ2−ϵ1
L2, then from (24), letting T = t0, we
immediately have that Ω(t) < ϵ2 if Ω(t0) ≤ ϵ1. Furthermore,suppose a⋆
≤ϵ3
d1ϵ2, then for all t ≥ t0
|si(t) − vl(t)| =
Nj=1
aij(t)sj(t − τ i
j (t)) − si(t)
≤
Nj=1
aij(t)Ω(t) ≤ d1a⋆Ω(t)
≤ ϵ3.
Proof of Theorem 3. If the condition C1 holds, sgn(vr) is equal toeither 1 or −1 for all t ∈ [t0, ∞). Let
k8 = k2vr,min − k3vr,max > 0. (79)
By (35), (39) and (79), we have that for all t ∈ [t0, ∞),
W ≤ −k1(V ′
xe)2− k8h−2
1 (V ′
ye)2− k7z2e . (80)
This shows that V (ρ(t), 12ϵ
24) ≤ W (t) ≤ W (t0) < ∞ for
all t ∈ [t0, ∞) (here, we use the shorthand ρ(t) to representρ(xe(t), ye(t))). As a consequence, ze is bounded and, by theproperty (b) of the potential function (see Section 2.2), there existsσ ∈ [0, 1
2ϵ24) such that ρ ∈ [0, σ ] for any t ∈ [t0, ∞). This implies
x2e (t) + y2e (t) < ϵ24 and that V ′
xe , V′ye are bounded over [t0, ∞). In
addition, there exists a positive constant k8 such that k8 < k8h−21 ,
hence (80) gives
W ≤ −k1(V ′
xe)2− k8(V ′
ye)2− k2z2e ≤ −k1(V ′
ρ)2ρ − k2z2e , (81)
where k1 = 2 ·mink1, k8. AsW is decreasing and bounded abovezero, W → c as t goes to ∞, where c is a nonnegative real. Toprove both ρ and ze converge to zero as t goes to infinity, it sufficesto show that c = 0. By contradiction, suppose c > 0. Then, by thecontinuity of V , there exists b > 0 such that either ρ > b or ze > bfor any t ∈ [t0, ∞). Following the property (c) of the artificialpotential function, (81) would imply that W is less than somenegative real for all t ∈ [t0, ∞), and hence W would eventuallybecome less than zero. This contradicts the nonnegativeness ofW . Finally, it follows from (32) that limt→∞ ρ = 0 implieslimt→∞ αφe = 0 which, together with limt→∞ ze = 0, giveslimt→∞ φe = 0.
If the condition C2 is satisfied, from (39) it follows that for anyt ∈ [t0, ∞), t
t0k1(V ′
xe)2+ k7z2e dτ = W (t0) − W (t)
−
t
t0k5vr sin(k6t)h−2
1 (V ′
ye)2dt
≤ W (t0) − W (t) +
t
t0k5|vr |dτ . (82)
Hence, W (t) ≤ W (t0) +
∞
t0k5|vr |dτ < ∞ for all t ∈ [t0, ∞),
which implies that there exists σ ∈ [0, 12ϵ
24) such that ρ ≤ σ for
all t ∈ [t0, ∞) and hence x2e (t) + y2e (t) < ϵ24 for all t ∈ [t0, ∞).
Note that (82) also shows that,∞
t0k1(V ′
xe)2+ k7z2e dt < ∞.
It is easy to verify that the integrand in the above inequality isuniformly continuous (Indeed, its derivative is bounded). Thus, bythe Barbalat Lemma (See, e.g., [39]),
limt→∞
V ′
xe = 0, limt→∞
ze = 0.
By the property (c) of the functionV , it is straightforward to deducelimt→∞ xe = 0 from the result limt→∞ V ′
xe = limt→∞ V ′ρxe = 0.
Now, from (33) and (34) we have that,
xe = −k1V ′
xe + ωye
= −k1V ′
xe +h−12 h−1
1 k5k6 cos(k6t)V ′yeye
1 − h−12 gV ′
xe
+ η, (83)
where η is a quantity converging to zero as t goes to infinity if vrandwr both tend to zero. Besides, it is easy to check that the secondterm in (83) is uniformly continuous over [t0, ∞). Applying Lemma2 in [42] to (83) gives that the second term in (83) tends to zero ast goes to infinity, which leads to
limt→∞
cos(k6t)V ′
yeye = 0. (84)
On the other hand, it can be obtained from (39) that W (t) −
k5|vr(t)| ≤ 0 for all t ≥ t0, which means W (t) − tt0k5|vr(τ )|dτ
is decreasing. On the other hand, from
∞
t0|vr(t)|dt < ∞ and the
nonnegativeness of W , we know W (t) − tt0k5|vr(τ )|dτ is lower
bounded. Hence W has a limit as t → ∞. But we know thatlimt→∞ xe = 0 and limt→∞ ze = 0. Thus ye must also have a limitas t → ∞, and from (84) we know that limt→∞ ye = 0. The resultlimt→∞ φe = 0 follows by the same reasoning as in the proof forthe previous case.
Proof of Lemma 1. Here we drop the vehicle index ‘‘i’’. By (4), wehave
(x′
r(s))2+ (y′
r(s))2
= (x′
q(s))2+ (y′
q(s))2+ (l2x(s) + l2y(s))(θ
′(s))2
− 2x′
q(s)lx(s) sin(θ(s))θ ′(s) − 2x′
q(s)ly(s) cos(θ(s))θ ′(s)
+ y′
q(s)lx(s) cos(θ(s))θ ′(s) − 2y′
q(s)ly(si) sin(θ(s))θ ′(s)
+ (l′x(s))2+ (l′y(s))
2+ 2l′x(s)
(x′
q(s))2 + (y′q(s))2
+ 2(lx(s)l′y(s) − l′x(s)ly(s))θ′(s).
Using (1), we obtain
(x′
r(s))2+ (y′
r(s))2
=
(x′
q(s))2 + (y′q(s))2 − ly(s)θ ′(s)
2+ l2x(s)(θ
′(s))2
+ (l′x(s))2+ (l′y(s))
2+ 2l′x(s)
(x′
q(s))2 + (y′q(s))2
+ 2(lx(s)l′y(s) − l′x(s)ly(s))θ′(s). (85)
If lx and ly are constants, from (85),
(x′
r(s))2+ (y′
r(s))2
=
(x′
q(s))2 + (y′q(s))2 − lyθ ′(s)
2+ l2x(θ
′(s))2. (86)
It is not difficult to see from
(x′q(s))2 + (y′
q(s))2 ≥ c > 0(Assumption 1) that the first term on the right hand side of (86)is lower bounded above zero when |θ ′(s)| < c8 for some c8 > 0.While if |θ ′(s)| ≥ c8, then by the condition (i) we have either|lxθ ′(s)| ≥ c4c8, or |
(x′
q(s))2 + (y′q(s))2 − lyθ ′(s)| = |(κ(s) −
ly)θ ′(s)| ≥ c5c8.Now, if lx(s) is constant but ly(s) can be varying, it follows from
(85) that
(x′
r(s))2+ (y′
r(s))2
=
(x′
q(s))2 + (y′q(s))2 − ly(s)θ ′(s)
2+l′y(s) + lxθ ′(s)
2,
1164 Q. Li, Z.-P. Jiang / Robotics and Autonomous Systems 60 (2012) 1149–1164
which can be easily checked lower bounded above zero if condition(ii) is satisfied.
The upper boundedness of
(x′r(s))2 + (y′
r(s))2 can be readilyjustified by Assumption 1 and the boundedness of x′
q(s), y′q(s),
x′′q(s), y
′′q(s) and lx(s), ly(s), l′x(s), l
′y(s). Lastly, the boundedness
of x′′r (s), y
′′r (s) can be shown by virtue of the boundedness of
x′′′q (s), y′′′
q (s) and l′′x (s), l′′y (s).
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Qin Li obtained his B.S. and M.S. degrees both fromBeihang University (formerly called Beijing University ofAeronautics & Astronautics), Beijing, in 2002 and 2005,and obtained his Ph.D. from the Polytechnic Institute ofNew York University, Brooklyn, in 2009. From September2009 to August 2011, he had been working as a Post-docresearcher at the Department of Mechanical Engineering,Eindhoven University of Technology. He is currentlya senior researcher at Statoil ASA. His main researchinterests include multi-agent systems, nonlinear control,and complex dynamical networks.
Zhong-Ping Jiang received his B.Sc. degree in mathemat-ics from the University of Wuhan, Wuhan, China, in 1988,his M.Sc. degree in statistics from the Université de Paris-sud, France, in 1989, and his Ph.D. in automatic control andmathematics from the École des Mines de Paris, France, in1993.
From 1993 to 1998, he held visiting researcher posi-tions with various institutions including INRIA (Sophia-Antipolis), France, the Australian National University, theUniversity of Sydney, and University of California. In Jan-uary 1999, he joined the Polytechnic Institute of New York
University at Brooklyn, New York, where he is currently a Professor. His main re-search interests include stability theory, the theory of robust and adaptive nonlinearcontrol, and their applications to underactuated mechanical systems, congestioncontrol, wireless networks, multi-agent systems and cognitive science.
Dr. Jiang has served as a Subject Editor for the International Journal of Robustand Nonlinear Control, and as an Associate Editor for Systems & Control Letters, IEEETransactions on Automatic Control and European Journal of Control. Dr. Jiang is a re-cipient of the prestigious Queen Elizabeth II Fellowship Award from the AustralianResearch Council, the CAREER Award from the US National Science Foundation, andthe Young Investigator Award from the NSF of China.
