Hierarchical-distributed optimized coordination of intersection traffic

0.79 MB·

by  Pavankumar Tallapragada

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Hierarchical-distributed optimized coordination
of intersection traffic
Pavankumar Tallapragada Jorge Corte´s
Abstract—This paper considers the problem of coordinating In contrast to traditional intersection management, networked
the vehicular traffic at an intersection and on the branches vehicle technologies allow us to coordinate the traffic not just
leadingtoitforminimizingacombinationoftotaltraveltimeand
within the intersection, but also by controlling the vehicles’
energy consumption. We propose a provably safe hierarchical-
6 behavior much before they arrive at the intersection. Such a
distributed solution to balance computational complexity and
1 optimalityofthesystemoperation.Inourdesign,acentralinter- paradigm offers the possibility of significantly reduced stop
0 sectionmanagercommunicateswithvehiclesheadingtowardsthe times and increased fuel efficiency, and is the subject of this
2 intersection, groups them into clusters (termed bubbles) as they paper.
n appear, and determines an optimal schedule of passage through Literature review: Much of the literature in the area of
a the intersection for each bubble. The vehicles in each bubble coordination-based intersection management focuses on colli-
J receivetheirscheduleandimplementlocaldistributedcontrolto
ensure system-wide inter-vehicular safety while respecting speed sionavoidanceofvehicleswithintheintersection.Supervisory
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andaccelerationlimits,conformingtotheassignedschedule,and intersection management (intervention only when required to

seeking to optimize their individual trajectories. Our analysis maintain safety by avoiding collisions) is explored using dis-
]
C rigorouslyestablishesthatthedifferentaspectsofthehierarchical crete event abstractions in [2] and reachable set computations
designoperateinconcertandthatthesafetyguaranteesprovided
O in[3],[4].Theworks[5],[6]andreferencesthereindescribea
by the proposed design are satisfied. We illustrate its execution
. in a suite of simulations and compare its performance to multiagent simulation approach in which, upon a reservation
h
traditionalsignal-basedcoordinationoverawiderangeofsystem request from a vehicle, an intersection manager accepts or
t
a parameters. rejects the reservation based on a simulation. Each vehicle
m
Index Terms—Intelligent transportation systems, hierarchical attempts to conform to its assigned reservation and, if this
[ and distributed control, optimized operation and scheduling, is predicted not to be possible at any time, the reservation

state-based intersection management, networked vehicles is canceled. [7] also uses a reservation-based system to
1
v schedule intersection crossing times and provides provably
6 I. INTRODUCTION safe maneuvers for vehicle following in a lane as well as
4 for crossing the intersection. [8], [9] use a method based
With rapidly growing urbanization and mobility needs of
2
on model predictive control to coordinate the intersection
0 people across the world, existing transportation systems are
crossingbyvehiclesandobtainsuboptimalsolutionstoalinear
0 in critical need of transformation. Apart from increased travel
quadraticoptimalcontrolproblem.[10]alsoproposesamodel
. times, current burdened transportation systems have the side
1
predictivecontrolapproachinwhichcollision-freeintersection
0 effects of increased pollution, increased energy consumption,
crossing by vehicles is achieved through a combination of
6 and degradation of people’s health, all of which have an
1 immeasurable cost on society. The complexity of the chal- hard no-collision constraints as well as a soft constraint in the
: form of a term measuring collision risk in the cost function.
v lenge requires a multi-pronged approach, one of which is the
In [11], a heuristic policy assigns priorities to the vehicles,
i developmentofnewtechnologies.Emergingtechnologiessuch
X while each vehicle applies a priority-preserving control and
asvehicle-to-vehicle(V2V)andvehicle-to-infrastructure(V2I)
r legacy vehicles platoon behind a computer-controlled car. In
a communication, and computer-controlled vehicles offer the
this context, we note that the ability to efficiently coordinate
opportunity to radically redesign our transportation systems,
diminishes as the vehicles get closer to the intersection. This
eliminating road accidents and traffic collisions and positively
is why here we take an expanded view of intersection man-
impacting safety, traveling ease, travel time, and energy con-
agement that looks at the coordinated control of the vehicles
sumption.
much before they arrive at the intersection. The methods
Aparticularlyusefulapplicationofthesetechnologiesisthe
above are not suited for this setup or would prove to be too
coordinationoftrafficatandnearintersectionsforasmoother
computationally costly in such scenarios. An example of the
(with reduced stop-and-go) and fuel-efficient traffic flow. An
expanded view of intersection management is [12], in which
intersectionmanagerwithknowledgeofthestateofthetraffic
a polling-systems approach is adopted to assign schedules,
could schedule the intersection crossings of the vehicles.
and then optimal trajectories for all vehicles are computed
With the assigned schedule, individual vehicles could further
sequentially in order. Such optimal trajectory computations
optimize their travel to the intersection in a fuel-efficient way.
are costly and depend on other vehicles’ detailed plans, and
A preliminary version of this work appeared as [1] at the 5th IFAC hence the system is not robust. Closer to this paper, the
WorkshoponDistributedEstimationandControlinNetworkedSystems. works [13], [14] describe a hierarchical setup, with a central
Pavankumar Tallapragada and Jorge Corte´s are with the Department of
coordinator verifying and assigning reservations, and with
MechanicalandAerospaceEngineering,UniversityofCalifornia,SanDiego
{ptallapragada,cortes}@ucsd.edu vehicles planning their trajectories locally to platoon and to
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meet the assigned schedule. The proposed solution is based II. PRELIMINARIES
on multiagent simulations and a reservation-based scheduling
Wepresentheresomebasicnotationandconceptsongraph
(with the evolution of the vehicles possibly forcing revisions
theory used throughout the paper.
to the schedule), both important differences with respect to Notation: We let R, R , Z, N, and N denote the set of
≥0 0
our approach. [15] is a recent survey of traffic control with
real, nonnegative real, integer, positive integer, and nonneg-
vehicular networks and provides other related references.
ative integer numbers, respectively. For a non-empty ordered
list S = {j ,...,j }, we let |S| denote the cardinality of S.
1 s
Statement of contributions: We propose a provably safe Further, S(i) denotes the ith element ji of S. Thus, S(|S|)
intersection management system aimed at optimizing a com- denotes the last element of S. For convenience, we also use
bination of cumulative travel time and fuel usage for all the the notation j ∈ S (j ∈/ S) to denote that j is (is not) an
vehicles. Our first contribution is the idea of coordinating the element of S. For two ordered lists S1 and S2, we let S1\S2
trafficattheintersectionandonthebranchesleadingtoitina denotetheorderedlistofelementsthatbelongtoS1 butnotto
unified, holistic way. The basic observation is that planning S2, while preserving the same order of S1. Given um ≤uM,
and controlling the vehicles from much before they arrive [u]uuMm denotesthenumberuloweranduppersaturatedbyum
at the intersection should lead to better overall coordination and uM respectively, i.e.,
and efficiency. Our second set of contributions is a multi- 
u , if u≤u ,
layered design that combines hierarchical and distributed con-  m m
tOroulrahnideraisrchaipcpalli-cdaibstlreibtuoteadwapidperoraacnhgeofofefrstraaffigcoocdonbdaitliaonncse. [u]uuMm (cid:44)uu, ,ifiuf ∈u≥[umu,u.M],
M M
betweencomputationalcomplexityofthesolutionandoptimal
Graphtheory: Wereviewbasicnotionsfollowingtheexpo-
operation.Theproposedsystemiscomposedofthreemainas-
sitionin[16],[17].AdigraphofordernisapairG=(V,E),
pects, each a contribution on its own: (i) clustering to identify
where V is a set with n elements called nodes and E is a set
vehicles that platoon before arriving at the intersection. We
of ordered pair of nodes called edges. A directed path is an
refer to such clusters of vehicles as bubbles. We use the term
orderedsequenceofnodessuchthatanyorderedpairofnodes
bubble, rather than platoon, to emphasize the dynamic, time-
appearing consecutively is an edge. A cycle is a directed path
varing nature of the cohesiveness of the group of vehicles as
thatstartsandendsatthesamenodeandcontainsnorepeated
theytraveltowardstheintersection.Withthisterminology,the
node except for the initial and the final one. A digraph is
bubblebecomesarigidandcohesivegroup(i.e.,aplatoon)by
acyclic if it has no cycles. A directed (or rooted) tree is an
the time they cross the intersection; (ii) a branch-and-bound
acyclic digraph with a node, called root, such that any other
scheduling algorithm that, using aggregate information about
nodecanbereachedbyoneandonlyonedirectedpathstarting
the bubbles, allows a central intersection manager to find the
at the root. If (i,j) is an edge of a tree, i is the parent of j,
optimal schedule of bubble passage; and (iii) a distributed
and j is the child of i. A node j is called a descendant of a
control algorithm for the vehicles at the local level. This
node i if there is a directed path from i to j. Given a tree,
controlpolicyensuresthatthevehiclesofeachbubbleplatoon
a subtree rooted at i is the tree that has i as its root and is
into a cohesive group when they cross the intersection and
composed by all its descendants in the original tree.
that each bubble conforms to the schedule prescribed by the
intersection manager, while guaranteeing system-wide safety
subject to speed limits and acceleration saturation. Addition- III. PROBLEMSTATEMENT
ally, each vehicle seeks to optimally control its trajectory Consider an intersection of length ∆ with four incoming
whenever safety is not immediately threatened. Our third traffic branches labeled by {1,2,3,4}, cf. Figure 1. For
and final contribution is the technical analysis leading to the
provable safety of our design. In contrast to computationally
intensive multiagent simulation-based methods, we provide
analytical guarantees on correctness, safety, and performance.
Further,theresultsprovidegoodintuitionandfundamentaland
reliableprinciplesforfuturedesigns.Wedoacknowledgethat
the development of analytical guarantees comes at the cost of
some conservatism in the design. We have performed a suite
of simulations comparing our approach to traditional signal-
basedcoordinationthatshowasignificantimprovementinthe
cumulative energy consumption for a wide range of traffic
densities and a more socially equitable distribution of cost.
However,thethroughputoftheintersectionissignificantlyless
in our approach than that of signal-based coordination except
for low densities of traffic. As a final note for the reader’s
Fig.1. Trafficnearanintersection.Blackdotsrepresentindividualvehicles,
sake, we have made every effort in the presentation to make
whichareclusteredandcontainedwithinbubbles,representedbygreyboxes.
thecomponentsofthepaperunderstandableeveniftheproofs ∆isthelengthoftheintersectionandthenumbers{1,2,3,4}arelabelsfor
of the technical results are skipped in a first reading. theincomingbranches.
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simplicity, we assume that (i) there is a single lane in each IV. OVERVIEWOFHIERARCHICALDISTRIBUTED
direction, (ii) all vehicles are identical with length L, (iii) SOLUTION
vehicles do not turn at the intersection, (iv) the intersection This section gives an outline of our hierarchical distributed
at any time may be used by vehicles from a single branch (v) solution to the problem stated in Section III. Our algorith-
there are no sources or sinks for vehicles along the branches mic solution combines optimized planning and scheduling of
- all new traffic appears at the beginning of the branches and groups of vehicles with local distributed control to ensure
must cross the intersection. We discuss later in Remark IV.1 safety and execute the plans. Its three distinct aspects are:
the extent to which these assumptions can be relaxed in our
(i) grouping the vehicles into clusters,
algorithmic solution.
(ii) scheduling the passage of the clusters through the inter-
The dynamics of vehicle j is a fully actuated second-order
section,
system,
(iii) local vehicular control to achieve and maintain cluster
cohesion, to avoid collisions, and to ensure the clusters
x˙vj(t)=vjv(t), (1a) meet the prescribed schedule.
v˙v(t)=uv(t), (1b) Eachoftheseaspectsiscoupledwiththeothertwo.Moreover,
j j
an overarching theme is the dynamic nature of the problem
where xv, vv ∈ R are the position (negative of the dis- due to the arrival and departure of vehicles. Any complete
j j
tance from the front of the vehicle to the beginning of the or partial solution has to be computed as new vehicles come
intersection) and velocity of the vehicle, respectively and in (event based) or at regular time intervals (time based). In
uv(t) ∈ [u ,u ], with u ≤ 0 ≤ u , is the input what follows, we provide a general description of the main
j m M m M
acceleration. We use the superscript v to emphasize that the ingredients of each aspect. At any given time t, we let ts be
state and control variables refer to individual vehicles. We the latest time prior to t at which the IM samples the state of
assume that each branch has a maximum speed limit that the trafficandsolvesthecorrespondingstaticschedulingproblem.
vehicles must respect. For the sake of easing the notation, we Aspect1–generationofbubbles: Theprimarymotivationto
assume that the speed limit on all branches is the same and clustervehiclesistoreducethenumberofindependententities
equals vM. Thus, for each vehicle j, vv(t) must belong to the that need to be considered in the (computationally expensive)
j
interval [0,vM] for all time t that the vehicle is in the system. schedule optimization problem. For instance, the maximum
number of clusters can be fixed according to the available
Each vehicle is equipped with vehicle-to-vehicle (V2V)
computational resources so that the scheduling problem re-
and vehicle-to-infrastructure (V2I) communication capabili-
mains tractable. At time t , the vehicles present in the four
ties. With V2I communication, the vehicles inform a central s
branches are grouped into N clusters. We let N denote the
intersection manager (IM) about their positionsand velocities k
numberofclustersonbranchk.Giventhepositioninformation
and receive from it commands such as prescribed time of
ofthevehiclesatt ,weusek-meansclusteringoneachbranch
arrivalattheintersection.WeassumetheIMhasthenecessary s
individually to identify the clusters. The relative positions of
communication and computing capabilities. We seek a design
thevehiclesofaclustermayvarysignificantlyoverthecourse
solution that aims to minimize a cost function C that models
of their travel and the vehicles may not be in the form of a
a combination of cumulative travel time and cumulative fuel
well-defined platoon at all times. Hence, we refer to a cluster
cost of the form
of vehicles as a bubble (shown as grey boxes in Figure 1).
(cid:90) Texit The defining characteristic of a bubble is that all the vehicles
(cid:88) j
C (cid:44) (WT +|uvj|)dt, (2) of a bubble cross the intersection together. The state of the ith
j tsjpawn bubble is given by the tuple
where j is the vehicle index, tspawn is the time at which ξi =(xi,vi,mi,τ¯iocc,Ii)∈R4×{1,2,3,4},
j
vehicle j ‘spawns’ into the problem domain and Texit is where x , v and m are, respectively, the position of the lead
j i i i
the time at which the vehicle exits the intersection, i.e., vehicle in the bubble, the velocity of the lead vehicle in the
xvj(Tjexit)=∆+L.TheweightWT setstherelativeimportance bubble, and the number of vehicles in the bubble. We denote
of travel time versus fuel cost. The vehicles over which the by τocc, the occupancy time of bubble i, which is the time
i
costissummedmaybechosenindifferentways-forexample duration for which the intersection is occupied by bubble i.
it may be over all vehicles that cross the intersection in a The quantity τ¯occ is an upper bound that can be guaranteed a
i
time period or it may be over a fixed number of vehicles. priori,andisafunctionofthebubblesizem andvariousother
i
The constraints in the problem arise from the speed limit, system parameters. The quantity I denotes which of the four
i
boundsonvehicleaccelerationanddeceleration,andthesafety incoming branches the bubble is on. Within each branch, we
requirements-whichrequireschedulingtheintersectioncross- require the order of the bubbles to remain constant during the
ing of the vehicles and maintenance of safe distance between bubbles’ travel (i.e., there is no passing allowed). To capture
the vehicles. Solving this problem at the level of individual theorderofthebubblesonabranch,wedefinethefunctionR,
vehicles is computationally expensive and not scalable. Thus, 
1, if I =I , x (t )<x (t ),
we aim to synthesize a solution that makes this problem  i q q s i s
R(i,q)(cid:44) (cid:64)i s.t. I =I , x (t )<x (t )<x (t ),
tractable to solve in real time and is applicable to a wide 1 i1 i q s i1 s i s
range of traffic scenarios. 0, otherwise.
4
Accordingtothisdefinition,R(i,q)=1ifandonlyifbubbles time of the bubble, τocc, is no more than τ¯occ. The scheduler
i i
i and q are on the same branch and bubble q is the immediate requires the quantity τ¯occ and other quantities such as earliest
i
follower of bubble i. We describe in detail the generation of and latest times of approach at the intersection for the bubble
bubbles and the algorithm to select the bubbles to schedule in that are functions of the initial conditions. All these quantities
SectionVbelow.Weimposealimitonthenumberofbubbles may be computed by the bubble and passed on to the IM or,
that are scheduled at any given time to N¯, even if the actual instead, the state of each car may be passed to the IM. We
number of bubbles in the system were greater, so as to keep assume that the control law at the vehicle level ensures that a
thecomputationalcostmanageable.However,inthealgorithm vehicledoesnotchangebubblesduringthecourseofitstravel
we describe in the sequel, each bubble is scheduled at least time. Thus, as far as the scheduling aspect is concerned, m
i
onceandsomebubblesmaybescheduledmorethanonce.We may be assumed constant in time. We describe in detail the
let t denote the latest time prior to t at which bubble i was local vehicular control component in Section VII below.
si
scheduled.
Remark IV.1. (Relaxation of assumptions). We discuss here
We index the vehicles in bubble i as (i,1),...,(i,m ),
i to what extent the assumptions made in Section III can be
where (i,1) refers to the lead vehicle in bubble i and so
relaxed in our proposed design. We make assumptions (ii)
on until (i,m ), the last vehicle in the bubble. We also find
i and (iv) only for the sake of simpler notation and ease of
it convenient for the label (i,0) to represent the last vehicle
exposition. Our algorithm can handle non-identical vehicles
(i(cid:48),m ) of the bubble i(cid:48) that precedes bubble i on the same
i(cid:48) with differing dimensions and differing acceleration limits,
branch or, if such bubble does not exist, we let (i,0) be an
thoughthosequantitiesneedtobeknown.Simultaneoususeof
imaginaryvehiclelocatedat∞.Wedroptheindexiwhenever
the intersection by vehicles on compatible branches/directions
there is no ambiguity with regard to the bubble.
is definitely possible in our framework and indeed makes the
Aspect 2 – scheduling of bubbles: The job of the scheduler
scheduling problem easier. We can relax assumption (v) if
is to prescribe to each bubble an approach time τ - the
i the sources or sinks are not close to the intersection with
time at which the ith bubble is to reach the beginning of
minorchangesinouralgorithmforbubblegeneration.Wecan
the intersection, i.e., x (τ ) = 0, so that no two different
i i avoidassumption(iii)andallowturningwithinourframework.
bubbles collide. In solving this problem, the scheduler has
However, the differing travel speeds when turning and going
to respect the order of bubbles on the same branch and take
straight affects the computation of the intersection occupancy
into account no-collision constraints between bubbles on two
time, which might make the design conservative. We believe
differentbranches.Thepreservationoftheorderofintersection
this conservativeness could be addressed by relaxing assump-
crossing by the bubbles on the same branch takes the form,
tion (i) and incorporating multiple lanes into the design. •
τ ≥τ +τ¯occ, if R(i,q)=1, (3a)
q i i
for i,q ∈{1,...,N}. Note that these constraints only ensure V. DYNAMICVEHICLECLUSTERING
that the passage of bubbles on a branch through the intersec- The primary motivation for clustering vehicles into bub-
tion occurs in the same order as they have arrived, but they bles is to reduce the computational burden on the scheduler.
donotnecessarilyexcludecollisionsfortheentiretraveltime. Consequently, we impose the upper bound N¯ on the number
Theintra-branchcollisionsareavoidedatalocallevelandwe of bubbles that the scheduler needs to consider at any given
accepttheresultingsub-optimality.Ontheotherhand,theno- instance. Further, as new vehicles arrive, they need to be as-
collisionconstraintbetweenbubblesontwodifferentbranches signed to new bubbles. In order to balance both requirements,
takes the form, we divide each branch into three zones, as shown in Figure 2:
staging zone (of length L ), mid zone (of length L ) and
τ ≥τ +τ¯occ OR τ ≥τ +τ¯occ, if I (cid:54)=I , (3b) s m
i q q q i i i q exit zone (of length L ). For each branch k ∈{1,2,3,4}, we
e
for i,q ∈{1,...,N}. The constraints (3b) make the schedul- let Zs, Zm and Ze be the set of positions on the branch k
k k k
ing problem combinatorial in nature because of the need to correspondingtothestaging,midandexitzones,respectively.
determine whether i or q goes first. Since the order on each
branch is to be preserved, the number of sub-problems is the
number of permutations of the multiset {I }N , i.e.,
k k=1
N! (cid:0)(cid:80)4 N (cid:1)!
= k=1 k ,
(cid:81)4 N ! (cid:81)4 N !
k=1 k k=1 k
where recall that N is the number of bubbles on branch k
k
and N is the total number of bubbles. We describe in detail Fig.2. Divisionofanincomingbranchintozones.
thealgorithmforoptimalschedulingofbubblesinSectionVI.
Aspect 3 – local vehicular control: The local vehicular The clustering into bubbles algorithm is executed
control has various equally relevant goals. The first goal is everyT unitsoftime.Ateachclusteringinstancet =sT ,
cs s cs
to avoid collisions within each bubble and among different s ∈ N , the vehicles in the staging zone that do not already
0
bubbles in the same branch. The second goal is for the local belong to a bubble are clustered. Thus, the choice Tcs < vLMs,
vehicular control to ensure that the bubble approaches the where recall that vM is the max speed limit, ensures that
intersection at the prescribed time τ and that the occupancy every vehicle belongs to a bubble before it leaves the staging
i
5
zone and enters the mid zone. We impose an upper bound Although we do not pursue here a systematic design of these
N¯ on the number of new bubbles that may be created on zonelengths,wecanidentifysomebasicobservationsoftheir
k
branch k at any clustering instance. At a clustering instance effect on clustering and scheduling. We envision these zone
t = sT , let nua denote the number of vehicles to be lengthstobeoftheorderofseveraltensofmeters.Thelength
s cs k
clustered in the staging zone of branch k. Then, the nua L of the staging zone has a direct effect on the time step of
k s
vehicles are clustered based on their position using the M - the periodic execution of clustering and scheduling as well as
k
means algorithm, with M = min{nua,N¯ }. Thus, the nua on the number of vehicles per bubble. The length L of the
k k k k m
on branch k are partitioned into M number of clusters or midzonehasaneffectonthelikelihoodofrevisingabubble’s
k
bubbles such that the sum of squares of the distances from scheduleonthenextiteration.Finally,thelengthL oftheexit
e
eachcartothecenterofitsbubbleisminimized,seee.g.,[18]. zonehasaneffectonthefeasibilityoftheschedulingproblem,
The clustering component in our design is modular and hence which we guarantee by assuming that L is large enough for
e
any clustering algorithm that is well suited may be used. a vehicle to come to a complete stop from a maximum speed
ThealgorithmalsomakessurethatnomorethanN¯ bubbles of vM in under a distance L . •
e
are passed to the IM manager for scheduling at any instance.
Remark V.2. (Re-clustering). The clustering into
This is achieved using two observations. First, previously
bubbles algorithm is just one method of defining bubbles
scheduled bubbles that have already entered the exit zone of
andselectingwhichonestoselect.Inthisalgorithm,avehicle
theirbrancharenolongerfedtotheIMforscheduling(i.e.,its
is assigned to a bubble only once and the vehicle is part
schedule is not modified any further). Second, if the number
of that bubble through out its travel. However, one could
of newly created bubbles and the previously created bubbles
yet to enter the exit zone exceeds N¯, then the algorithm pops implement a strategy which re-clusters all vehicles in the
staging and mid zones so that vehicles may be reassigned to
out the required number of bubbles from the top of the list
a different bubble, bubbles may be merged or split as needed,
of bubbles previously scheduled (corresponding to the ones
and so on. Such an algorithm would also allow sources and
closer to their respective exit zones). We present the precise
sinks on the branch such as smaller streets, homes, and retail.
description of the clustering into bubbles algorithm in
•
Algorithm 1.
Algorithm 1: clustering into bubbles at sT VI. SCHEDULINGOFBUBBLES
cs
Input: Lp,τpmin This section describes the scheduling algorithm employed
{Ordered list of bubbles scheduled at (s−1)Tcs and bytheintersectionmanager(IM)todecidetheorderofpassage
earliest approach time used in scheduling them} through the intersection of the bubbles in L provided by
1: L←Lp\{j∈L:Ij =k ∧ xj ∈/Zks∪Zkm} the clustering algorithm. The scheduling algorithm is also
{remove bubbles that are not completely within
the staging and the mid zones} executed every Tcs units of time. In this section, we let L be
2: fork=1to4do theset{1,...,N},whereN =|L|,withoutlossofgenerality.
3: N¯k {max new bubbles on branch k}
4: Mk←min{nuka,N¯k} {# new bubbles on branch k}
5: Clusternewvehiclesonbranchk usingMk-meansalgorithm A. Cost function and constraints
6: endfor
4
7: M←(cid:88)Mk Inourapproach,theIMschedulesbubblesasawholeusing
k=1 an abstraction of the vehicle dynamics and the cost function.
8: ifM+|L|>N¯ then
First, regarding the vehicle dynamics, we note that the inter-
9: RemovefirstM+|L|−N¯ bubblesfromL {Ensure only N¯
bubbles provided to scheduler by dropping the vehicle approach times at the intersection and the resulting
earliest bubbles in previous schedule} occupancy time of a bubble is a degree of freedom. However,
10: endif
we have made the alternative choice of not considering it
11: AppendnewbubblestoL
12: τmin←max(cid:0){τpmin}∪{τi+τ¯iocc:i∈Lp\L}(cid:1) as such in the scheduling algorithm, and instead only use
{earliest approach time for the bubbles in L} an upper bound on the occupancy time τ¯occ (that the local
Output: L,τmin vehicular control component can guaranteei) appearing in the
constraints (3). Second, regarding the cost function, we ab-
The algorithm takes in the list of bubbles Lp scheduled stract the fuel cost for the vehicles in a bubble i into a single
on the last iteration and an earliest approach time τmin used function F that depends only on the average velocity of the
p i
when scheduling it. The output is a list of bubbles L to be bubble i (lead vehicle in the bubble) for t ∈ [t ,t + τ ],
s s i
scheduled and the earliest approach time τmin for them. Note wheret =sT isthetimeatwhichtheschedulingalgorithm
s cs
from step 12 of Algorithm 1 that τmin is an upper bound is executed. Thus, the scheduling algorithm minimizes the
on the time by which all the bubbles not in the L list are following simplified cost function C (cid:44) C where C for a
L P
guaranteed to cross the intersection. Thus, when scheduling given list of bubbles P is
L, the scheduler imposes the constraint that the bubbles in L
(cid:88)
approach the intersection no earlier than τmin. CP (cid:44) mi(WTτi+Fi(v¯i))
i∈P
Remark V.1. (Effect of zone lengths on clustering and
scheduling). The lengths of the three zones illustrated in =(cid:88)mi(cid:16)WTdv¯i +Fi(v¯i)(cid:17)(cid:44)(cid:88)φi(v¯i), (4)
Figure 2 directly affect the resulting traffic coordination. i∈P i i∈P
6
where v¯ is the average velocity of the lead vehicle in bubble B. Optimalbubbleaveragevelocityforfixedorderofpassage
i
i for t ∈ [ts,ts+τi], i.e., v¯i = dτii, where di (cid:44) −xi(ts). The Hereweaddresstheproblemofdetermining,givenadesired
optimization variables are v¯ for each bubble i∈L.
i order of bubble passage through the intersection, the optimal
Note that in the cost function C, the functions F could, in
i average velocities of the bubbles and the associated optimal
general, depend on initial conditions modeled as parameters
cost. For this purpose, define an order of the approach times
- such as the distance d to reach the intersection. The cost
i of the bubbles as a permutation, P, of the integers from 1 to
function (4) models a combination of cumulative travel time |P|≤N. We use P(i) to denote the ith element in the order,
and total fuel usage. Motivated by the fact that fuel efficiency
withthebubbleP(1)passingthroughtheintersectionfirstand
is typically an increasing function of vehicle speed for speeds
so on. We use σ (i) to denote the position of bubble i in the
P
underthelimitsenforcedonmostroadswithintersections,we
order P. Clearly, for a permutation to respect the intra-branch
maketheassumptionthat,foreachi∈L,F :[0,vM](cid:55)→R
i >0 orders, σ (i)<σ (q) if R(i,q)=1. Given P respecting the
P P
is monotonically decreasing.
intra-branch orders, the bubble velocity optimization
Regardingtheconstraints,conditionsonthetraveltimescan
algorithm, formally described in Algorithm 2, finds a so-
be re-expressed as conditions on average velocities as
lution to the optimization of C under the constraints (6),
P
τ ≥τ +τ¯occ ⇐⇒ di ≥ dq +τ¯occ v¯i ∈[v¯im,v¯iM], and with order P.
i q q v¯ v¯ q
i q
d τ¯occ Algorithm 2: bubble velocity optimization
⇐⇒ v¯ ≥c v¯ +b v¯ v¯, c = q, b = q . (5)
q qi i qi q i qi di qi di Input: OrderP
1: C←0
Thus, we re-express the no-collision constraints (3) as
2: forh=1to|P|do
3: i←P(h) {bubble i is in position h in P}
v¯ ≥c v¯ +b v¯v¯ , if R(i,q)=1, (6a)
i iq q iq i q 4: ifh=1then
v¯ ≥c v¯ +b v¯ v¯ OR v¯ ≥c v¯ +b v¯v¯ , if I (cid:54)=I . 5: v¯P ←v¯M
q qi i qi q i i iq q iq i q i q i i
6: else
(6b)
7: q←P(h−1) {bubble q is in position h−1 in
In addition, we also need to ensure that the scheduling at P}
instance sT of the bubbles in L does not conflict with 8: v¯P ←min{v¯M, v¯qP }
cs i i cqi+bqiv¯qP
the ones that have been previously scheduled. Formally, this 9: endif {v¯P is the optimizer for bubble i}
i
corresponds to having the time τi to reach the intersection for 10: C←C+φi(v¯iP) {update cost}
bubble i be no less than τmin (cf. step 12 of Algorithm 1). 11: endfor
Equivalently, we require
The following result shows that, for an order that respects
d
v¯ ≤ i . (6c) the intra-branch order, the algorithm finds the average veloci-
i τmin
ties that optimize the cost function C .
P
Notethattheschedulingproblemiscombinatorialinnature
due to the no-collision constraints (6b). Thus, even though Lemma VI.1. (bubble velocity optimization algo-
the cost function C is simple and the optimization variables rithm optimizes the schedule given an overall order that
are the average velocities v¯, we believe this formulation pro- respects the intra-branch orders). For each i ∈ {1,...,N},
i
assume the fuel cost function F is monotonically decreasing.
vides a good balance between usefulness and computational i
Let P, with |P|≤N, be an order respecting the intra-branch
tractability. Further, the local vehicular control we present in
orders and denote by v¯P =(v¯P,...,v¯P) and C the output of
Section VII seeks an optimal control profile to achieve the 1 N
Algorithm 2. Then, v¯P and C are, respectively, the minimizer
prescribed average velocity for the bubble, which justifies the
restrictiontov¯ astheoptimizationvariablesinthescheduling andtheminimumcostoftheoptimizationproblemwiththecost
i
aspect. Thus our proposed solution, although sub-optimal, is function as CP (4) under the constraints (6), v¯i ∈[v¯im,v¯iM].
still principled.
Proof: Given the order P, the constraints (6) reduce to
We next describe our solution to the scheduling problem
v¯
cstornasinisttsin(g6)ofanmdinv¯im∈izin[gv¯mC,v¯M=].CTLheinlo(w4)erunv¯dmer≥the0caonnd- cqi+qbqiv¯q ≥v¯i
i i i i
upper v¯iM ≤vM limits on the average velocity depend on the where q =P(h−1), i=P(h) and h∈{2,...,N}. The left-
initialconditionsofthevehiclesanddesiredspeedlimits.The hand side of the inequality is an increasing function of v¯ .
j
quantitiesv¯im andv¯iM areinverselyrelatedtothelatesttimeof Further since Fi is a monotonically decreasing function for
approach and the earliest time of approach at the intersection each i, v¯P takes the maximum possible value. The algorithm
i
for bubble i, respectively. The computation of these quantities computes the components of v¯P iteratively and the result
is described in Section VII-A. Similarly, the upper bound τ¯iocc follows.
ontheoccupancytimesmaybecomputedasinSectionVII-C.
In the first part of our solution to the scheduling problem, we
C. Optimal ordering via branch-and-bound
determinetheoptimalscheduleandoptimalcostgivenafixed
orderofbubblepassagethroughtheintersection.Inthesecond We propose a branch-and-bound algorithm to solve the
part,weuseabranch-and-boundalgorithmtofindtheoptimal optimalschedulingproblem.Westartbyprovidinganinformal
order and schedule. description.
7
Informaldescription:Abranch-and-boundalgorithm Thislowerboundispreciselywhatisrequiredtoimplementa
consists of a systematic enumeration of the set of branch-and-bound algorithm to find the optimal schedule for
candidate solutions as a rooted tree, with the full the bubbles.
set at the root. The algorithm explores branches Specifically,thebranch-and-boundalgorithmstartsbypick-
of the tree, which represent subsets of the set of ing an arbitrary candidate order and computing the cost for it,
candidate solutions. Before enumerating the candi- usingthebubble velocity optimizationalgorithm,and
date solutions of a branch, the branch is checked storing the two as the current best solution and cost. Then,
againstupperboundsontheoptimalsolution,andis starting at the root node of the tree of all feasible orders,
discarded if it is determined that it cannot produce the algorithm searches (e.g., using depth-first or breadth-first
a better solution than the best one found so far. search) for an optimal solution. If at any time a leaf node,
which corresponds to a fully determined order, is reached
Weformallyspecifyeachofthecomponentsinthisdescrip-
and its cost is better than the current best, then the current
tion next, starting with the rooted tree. We let P denote any
best solution and cost are updated. For any other node P
ordered list of up to length N, with non-repeating numbers
in the tree, (7) provides a lower bound CP on the cost of
drawn from {1,...,N}, and preserving the individual branch
all the orders represented by the node P. If CP is greater
orders. With this notation, the empty list P = ∅ denotes the
than the current best known cost, then the subtree P is
root of the tree, representing all feasible orders. Similarly,
discarded.Thisprocesscontinuesuntilthealgorithmfindsthe
P =(i ,...,i ) denotes the subtree of all the feasible orders
1 h
optimal solution. We refer to this process as the schedule
in which bubble i crosses the intersection first, i second,
1 2
and so on, until bubble i is the hth to cross, with the order optimization algorithm.
h
of the remaining bubbles undetermined.
VII. LOCALVEHICULARCONTROL
Our next step is to provide a way to determine a lower
The local vehicular control component of our hierarchical-
bound on the achievable optimal value of any given branch.
distributed coordination approach involves two main tasks:
This follows from the observation that (i) the execution of
(i) compute, for each bubble i, the lower v¯m and upper v¯M
the bubble velocity optimization algorithm finds the i i
averagevelocitybounds,andtheupperboundontheintersec-
optimalvalueoftheaveragevelocityabubblegiventheorder
tion occupancy time τ¯occ that are provided to the scheduler;
of all the bubbles preceding it, but (ii) one can compute an i
and (ii) control the vehicles ensuring no collisions and that
upper bound for the optimal value even if only part of the
all the vehicles of bubble i cross the intersection within
order of bubbles preceding it is known. The description in
the time interval [τ ,τ + τ¯occ] prescribed by the scheduler.
Algorithm 3 of this procedure, termed bounding optimal i i i
The successful execution of each of these tasks requires an
bubble velocity algorithm, relies on four ordered lists,
understanding of the vehicle dynamics and the desired safety
termed queues, one for each branch. The queue for branch
constraints and the effect of each on the other. The following
notion of safe-following distance is particularly useful in our
Algorithm 3: bounding optimal bubble velocity
forthcoming developments.
1: l←P(|P|) {l is last bubble in P}
2: Compute v¯P using bubble velocity optimization with Definition VII.1. (Safe-following distance). The maximum
l
inputP braking maneuver (MBM) of a vehicle is a control action
3: fork=1to4do that sets its acceleration to u until the vehicle comes to
4: Qk←Qk\P {pop-out P from Qk} a stop, at which point its accemleration is set to 0 thereafter.
5: ifQk(cid:54)=∅then
6: i←Qk(1) {i is first of remaining bubbles in Let j −1 and j be the indices of two vehicles on the same
Qk} branch, with vehicle j immediately following j −1. We say
7: HiP ←min{v¯iM,cli+v¯blPliv¯lP} a quantity D(vjv−1(t),vjv(t)) is a safe-following distance at
8: fors=2to|Qk|do timetforthepairofvehiclesj−1andj ifxv (t)−xv(t)≥
9: i←Qk(s) D(vv (t),vv(t)) and, if each of the two vj−eh1icles wjere to
10: q←Qk(s−1) j−1 j
11: HP ←min{v¯M, HqP } perform the MBM, then the two vehicles would be safely
12: endfior i cqi+bqiHqP separated, xvj−1−L≥xvj (recall that L is the vehicle length)
until they come to a complete stop. •
13: endif
14: endfor
According to this definition, a safe-following distance is
not uniquely defined, which in fact provides a certain leeway
k, Q = (i ,...,i ), is initialized to the list of all
k k,1 k,Nk in designing the local vehicle control. The following result
the bubbles on branch k in their order of arrival (thus
identifies a specific safe-following distance.
R(i ,i ) = 1 for all q ∈ {1,...,N −1}). We denote
k,q k,q+1 k
by HP the upper bound on the average velocity v¯ of bubble Lemma VII.2. (Safe-following distance as a function of
i i
vehicle velocities). Let j−1 and j be a pair of vehicles, with
iobtainedbyAlgorithm3giventhatanon-emptyP precedes
j following j−1. Then, the continuous function D defined by
it. This allows us to lower bound the optimal cost for any
order in the subtree P in terms of v¯P and HP as follows,
i i D(vv (t),vv(t))=
j−1 j
CP (cid:44)(cid:88)φi(v¯iP)+ (cid:88) φi(HiP). (7) L+max(cid:110)0,−21u (cid:0)(vjv(t))2−(vjv−1(t))2(cid:1)(cid:111), (8)
i∈P i∈L\P m
8
provides a safe-following distance at time t for the pair of τe −(j−1)Tnom.Hence,wedefineearliesttimeofapproach
i,j
vehicles j−1 and j. for the bubble i, τe as
i
Proof:Ifavehiclej withdynamics(1)weretodecelerate τe (cid:44)max{τe −(j−1)Tnom :j ∈{1,...,m }}, (9)
i i,j i
at the maximum rate possible (acceleration equal to uv <0)
f−rovmv(tc)u/ruren,ttthimenetuntilitcomestoacompletestopatmtsjtop = iamndumletdev¯ciMele=rati−onxτiie(ytsie)l.dAtnhaelolgaoteusst ctiommepuotfataiopnpsrowacithh τmlaxo-f
j m i
bubble i, possibly with τl =∞, and the corresponding lower
xv(tstop)=xv(t)+ (vjv(t))2. bound v¯im ≥0 for the avierage velocity. Hence, the values we
j j j −2u obtain inthis wayfor v¯m and v¯M are, respectively, larger and
m i i
smallerthantheoneswewouldhaveobtainedifweonlytook
If vv(t) ≥ vv (t) ≥ 0, then the safe-following distance is
j j−1 into account the lead vehicle of the bubble.
found by setting
For a given upper bound on the occupancy time and the
xv (tstop)−xv(tstop)≥L. sets of v¯im and v¯iM for i ∈ L, a feasible schedule might
j−1 j−1 j j
not always exist. Thus, to guarantee the feasibility of the
Ifontheotherhandvv (t)≥vv(t)≥0,thenthevehiclesare scheduling problem in a simple fashion, we assume that the
j−1 j
infactclosestattimetandtheconditionxv (t)−xv(t)≥L exitzonelengthL islargeenough.Specifically,wemakethe
j−1 j e
is sufficient to ensure subsequent safety. Hence (8) provides a following observation.
safe following distance.
Lemma VII.4. (Existence of a feasible schedule). If the exit
Remark VII.3. (Monotonicity properties of D). If the first (vM)2 (νnom)2
zone length, L ≥ + , then there always exists
argument of the function D is fixed, then it is monotonically e −2um 2uM
a feasible schedule with which each vehicle is able to enter
non-decreasing. On the other hand, if the second argument is
the intersection with a speed of at least νnom.
fixed then the function is monotonically non-increasing. •
Proof: Recall, that a schedule to a bubble is assigned
when all the vehicles in the bubble are still in the staging or
A. Bounds on average bubble velocity
the mid zones. Clearly, the condition on L implies that any
e
Recall that v¯ is the average velocity of the lead vehicle of vehicle in the staging zone or the mid zone (xv ≤ −L ) can
i j e
bubbleifromt anduntiltheleadvehicleissupposedtoreach come to a complete stop and then accelerate to a speed of at
s
thebeginningoftheintersectionatτ .Thus,itwouldseemthat least νnom before arriving at the beginning of the intersection
i
computing lower and upper bounds on the achievable average (xv =0).
j
velocity of the lead vehicle in the bubble is sufficient to
determinev¯M andv¯m.However,ignoringtheinitialconditions
i i B. Vehicle controller design
of the other vehicles in the bubble in the computation of v¯M
i
and v¯m poses the risk of lengthening the guaranteed upper The scheduler prescribes for each bubble a time at which
i
bound τ¯occ on the occupancy time. The reasoning for this is the vehicles in the bubble may start to cross the intersection.
i
The local vehicular control must ensure that the vehicles
better explained in terms of earliest times of approach at the
of bubble i start and finish crossing the intersection within
intersection of the vehicles.
In bubble i, we let τe be the earliest time vehicle (i,j) the time interval [τi,τi + τ¯iocc] while respecting the safety
i,j
constraints (8). In this section, we describe an algorithm to
can reach the intersection ignoring the other vehicles on the
achieve this task. The algorithm has three main parts: (i)
branch. Letting t =s T be the time at which bubble i was
last scheduled, thsei quanitictsy τe −t is then the time it takes an uncoupled controller ensuring that the vehicle arrives at
xv toreach0fromxv (t )if,ojrthesitrajectorywithmaximum the intersection at a designated time if the presence of all
i,j i,j si other vehicles is ignored. This controller is applied when
acceleration until vv = vM and zero acceleration thereafter.
i,j the preceding vehicle is sufficiently far in front, (ii) a safe-
Thus,weseethatifτe forsomej >1issignificantlygreater
i,j following controller ensuring that the vehicle follows the
than τe then the vehicle (i,1) has to slow down to approach
i,1 preceding vehicle safely when the latter is not sufficiently far
the intersection at a time later than τe so that the guaranteed
i,1 infront;and(iii)aruletoswitchbetweenthetwocontrollers.
upper bound τ¯occ on the occupancy time is small enough.
i 1) Uncoupledcontroller: Foreachvehiclej ∈{1,...,m }
Thus, we propose the following alternative solution. As- i
in bubble i, we define,
suming a nominal speed νnom for vehicles when entering the
intersection, we set Dnom (cid:44) D(νnom,vM), which has the τ (cid:44)τ +(j−1)Tnom. (10)
i,j i
connotation of a safe inter-vehicle distance given a vehicle is
traveling at the maximum allowed speed vM and the vehicle Given the constraints that the scheduler takes into account,
preceding it traveling at a speed greater than or equal to νnom. we have τi ∈ [τie,τil]. This, together with (9), implies that
Then, we also define Tnom (cid:44) Dnom/νnom as the nominal τi,j ∈[τie,j,τil,j]. Now, let
inter-vehicle approach time. With this nominal inter-vehicle (t,xv ,vv )(cid:55)→g (τ ,t,xv ,vv )
approachtimesofvehiclesinabubbleweseethattheearliest i,j i,j uc i,j i,j i,j
time of approach for vehicle (i,j) forces the earliest time be a feedback controller that ensures xv (τ ) = 0 for the
i,j i,j
of approach of bubble i, i.e. vehicle (i,1), is no less than dynamics (1) starting from the current state (xv (t),vv (t))
i,j i,j
9
at time t (assuming feasibility), respecting the control and preceding vehicles and no control exists to ensure Ta =τ
i,j i,j
velocityconstraints,butnotnecessarilytheinter-vehiclesafety alongwiththeotherconstraints.Additionally,fort>Ta ,i.e.,
i,j
constraints. We refer to it as the uncoupled controller. Such after the vehicle enters the intersection, the optimal controller
a controller exists for each vehicle at least at t = t , where isnotwelldefinedanddoesnotexist.Asashorthandnotation,
si
t =s T is the time at which bubble i was scheduled, due we use ∃F (respectively (cid:64)F ) to denote the existence
si i cs i,k i,k
to the fact that τ ∈ [τe ,τl ]. Here, we take as g the (respectively, lack thereof) of an optimal control g . In order
i,j i,j i,j uc uc
optimal feedback controller that generates velocity profiles as for the control g to be well defined at all times, we let
uc
shown in Figure 3 obtained by optimizing
g (τ ,t,xv ,vv )(cid:44)u , if (cid:64)F .
uc i,k i,k i,k M i,k
(cid:90) τj
|uv(s)|ds 2) Controllerforsafefollowing: Asmentionedearlier,this
j
t controllerisappliedonlywhenavehicleissufficientlycloseto
with optimization variables a , a (the areas of the indicated the vehicle preceding it. Besides maintaining a safe-following
1 2
triangles), vv(τ ), νl and νu, where we have dropped the distance, the controller must also ensure that the resulting
j j
bubble index i. The constraints are νl ∈ [0,vv(t)], νu ∈ evolution of the vehicles in the bubble i is such that the
j
[vjv(t),vM], vjv(τi,k) ∈ [νnom,vM], a1,a2 ≥ 0 and that the occupancytimeisnomorethanτ¯iocc.Here,wepresentadesign
total area under the curve must be equal to −xv(t). The toachievethesegoals.Forapairofvehiclesj−1andj,with
j
feedback controller may be found by tabulating the optimal j following j−1, we define the safety ratio as
control solution. xv (t)−xv(t)
σ (t)(cid:44) j−1 j , (11)
j D(vv (t),vv(t))
j−1 j
which is the ratio of the actual inter-vehicle distance to the
safe-following distance. Hence, we would like to maintain
this quantity above 1 at all times. Notice from (8) that if
vv (t) > vv(t), then σ increases and safety is guaranteed.
j−1 j j
Thus, it is sufficient to design a controller that ensures safe
following when vv(t) ≥ vv (t). For vehicle j, we denote
j j−1
(a) ζ (cid:44)(vv ,vv,σ ). Define the unsaturated controller g by
j j−1 j j us
g (ζ ,uv )(cid:44)
us j j−1
(cid:40)
uv , if vv =0,
j−1 j
(cid:16)vjvv−jv1 (cid:16)1+σj−uvju−m1(cid:17)−1(cid:17)(cid:16)−σujm(cid:17), if vjv >0.
Therationalebehindthisdefinitionisasfollows.Asmentioned
above, it is sufficient to design a controller that ensures safe
(b) following when vv(t) ≥ vv (t). Thus, if vv = 0 then we
j j−1 j
Fig.3. Candidatevelocityprofilestoobtainguc,whichtakesform(a)or(b) need to consider only the case of vv = 0. In this case, the
dependingonthevelocityvjv(t),νnom,vM,τjandthedistancetogo−xvj(t). definition of g ensures that the vje−h1icle j stays at rest as
us
long as vehicle j−1 is at rest and starts moving only when
j−1 starts moving again. Further, since the relative velocity
RemarkVII.5. (Optimalityofthecontroller).Assumingthere
and acceleration in this case would be zero, we see that σ
exists a feasible controller that ensures the vehicle (i,j) j
stays constant. As we see more thoroughly in the sequel, if
approaches the intersection at τ with a minimum velocity
i,j
of νnom, ignoring any other vehicles on the branch and given the vehicle is moving, vjv >0, then guc ensures that σj stays
the current time t and the vehicle state (xv (t),vv (t)), then constantandthusensuringsafety.However,inthesecondcase,
i,j i,j g might cause vv to exceed vM. Further, we would like
there exists an optimal solution with piecewise-constant-rate us j
thevehicletocontinueusingtheoptimaluncoupledcontroller
velocity profiles as shown in Figure 3. We can see this
if it does not affect the safety by decreasing σ . These
statement to be true by observing that in a given time τ −t, j
j
considerations motivate our definition of the safe-following
theminimumandmaximumtraveldistancesareobtainedwith
controller as
velocity profiles belonging to the family depicted in Figure 3,
and that every other intermediate travel distance is obtained g (t,ζ ,uv )(cid:44)
by a continuous variation of the velocity profiles within the sf j j−1
min{g (τ ,t,xv,vv),g (ζ ,uv )}. (12)
family. • uc j j j us j j−1
3) local vehicular controller: Here, we design the
Note that the control g assumes the presence of no other
uc
local vehicle controller by specifying a rule to switch be-
vehicles on the branch. Thus, the actual approach time, Ta ,
i,j tween the uncoupled controller g and the safe-following
of the vehicle (i,j) may be later than τ . At time t , when uc
i,j si controller g . To make precise whether two vehicles are
the bubble is scheduled, an optimal control g does exist sf
uc sufficiently far from each other, we introduce the coupling
for each of its vehicles because of the way the times τ are
i,j set C defined by
defined in (10). However, at a future time t, such a feasible s
g might not exist because the vehicle is slowed down by C (cid:44){(v ,v ,σ):v ≥v and σ ∈[1,σ ]}, (13)
uc s 1 2 2 1 0
10
with σ > 1 a design parameter. Intuitively, if ζ ∈ C , then statesthattherelativeacceleration,andhencealsotherelative
0 j s
vehicle j is going at least as fast as the vehicle in front of velocity,stayszero.Finally,claim(iv)isanecessarycondition
it, and their safety ratio is close to 1. With this in mind, we on the velocity of vehicle j − 1 for g and the saturated
us
define the local vehicular controller for vehicle j, [g ]0 to differ.
us um
 The following result states that if at any instant in time the
g , if ζ ∈/ C , vv <vM,
uv(t)=[guucc]0um, if ζjj ∈/ Css, vjjv =vM, (14) oslpotwimedaldcoowntnroblylerprdeoceesdninogtevxeihsitc(lebse)c,atuhseenthaevveehhiciclelenhoatsinbetehne
j g[gsf,]0 , iiff ζζj ∈∈CCs,, vvjvv <=vvMM,. coupling set moves at the maximum speed.
sf um j s j Lemma VII.7. (Vehicle exits the coupling set at maximum
Note that [g ]0 (cid:54)= g only if (cid:64)F . This controller has the speed if the optimal controller does not exist). Let t1 be any
uc um uc j time such that ζ (t )∈C and ζ (t)∈/ C for t∈(t ,t +δ)
vehicleuse thesafe-following controller whenin thecoupling j 1 s j s 1 1
for some δ > 0. If (cid:64)F at time t , then vv(t) = vM for all
set, and the uncoupled controller otherwise. The following j 1 j
t∈[t ,t +δ).
result describes some features of the dynamical behavior of 1 1
vehicles under (14) when in the coupling set. Proof: Under the hypotheses of the result, and as a
consequence of Lemma VII.6(iii), the only way vv(t ) =
Lemma VII.6. (Vehicle behavior in the coupling set). For j 1
some t, let ζj(t) ∈ Cs and uvj−1(t) ∈ [um,uM]. Then, the vHjvo−w1e(tv1e)r,ibsypaossssuibmleptiisonif(cid:64)gFsfa=t tigmuec t<.uTMhusa,titt1f,olil.oew.,s∃tFhajt.
following hold: j 1
vv(t )>vv (t ). By definition of t , we then conclude that
(i) g (ζ ,uv )∈[u ,u ], j 1 j−1 1 1
us j j−1 m M σ (t ) = σ . Next, at t , since (cid:64)F it means g = u and
(ii) If vv < vM and g (t,ζ ,uv ) = g (ζ ,uv ) or if j 1 0 1 j uc M
j sf j j−1 us j j−1 thus g = g . Then, from (ii), we see that vv(t ) < vM is
vv = vM and g (t,ζ ,uv ) =[g (t,ζ ,uv )]0 = sf us j 1
gj (ζ ,uv ), thsefn σ˙ j=j0−,1 sf j j−1 um not possible and that in fact vjv(t1) = vM and gsf = gus >
us j j−1 j [g ]0 =0. During the interval (t ,t +δ), we see from the
(iii) If vjv = vjv−1 ≥ 0 and gsf(t,ζj,uvj−1)= gus(ζj,uvj−1), sescfonudmcase of (14) that uv =[g ]10 1=[u ]0 =0, which
then σ˙j =0 and uvj =uvj−1, proves the result. j uc um M um
(iv) If vv = vM, then g (ζ ,uv ) ≥ [g (ζ ,uv )]0 =
j us j j−1 us j j−1 um
0 only if
C. Upper bound on guaranteed occupancy time
−u vM
vv ≥vv (cid:44) m . The last element of the design is the upper bound on the
j−1 −u +σ u
m 0 M guaranteed occupancy time for a bubble. To obtain this, we
Proof:Forthesakeofconciseness,wedropthearguments first upper bound the inter-approach times of vehicles in a
of the functions wherever it causes no confusion. given bubble at the beginning of the intersection.
(i)Forvv =0,theclaimreadilyfollowsfromthedefinition
j PropositionVII.8. (Upperboundontheinter-approachtimes
of g . For fixed σ ≥1, vv ≥vv ≥0 and vv >0, we see
us j j j−1 j of vehicles in a bubble at the intersection). For any bubble i
that g is maximized and minimized when uv = u and
us j−1 M and any vehicle j ∈ {2,...,m }, if vv (Ta ) ≥ νnom,
uv =u ,respectively.Theresultthenfollowsbyobserving, i i,j−1 i,j−1
j−1 m then vv (Ta )≥νnom and Ta −Ta is upper bounded by
after some computations, that g (ζ ,u ) − u ≤ 0 and i,j i,j i,j i,j−1
us j M M
gus(ζj,um)−um ≥0. (cid:40)σ Tnom, if vv ≥νnom,
(ii) and (iii) From (11) observe that Tiat (cid:44) 0
max{σ Tnom,Tfol(vv)}, if vv <νnom,
0
vv −vv−σ D˙(vv (t),vv(t))
σ˙ = j−1 j j j−1 j where vv is defined in Lemma VII.6(iv) and
j D(vv (t),vv(t))
j−1 j σ D(v,vM)−D (v)
vv −vv− σj (vvuv−vv uv ) Tfol(v)(cid:44) 0 g ,
= j−1 j −um j j j−1 j−1 vM
D(vv (t),vv(t)) (cid:16)νnom−v(cid:17) (cid:16)νnom+v(cid:17)(cid:16)νnom−v(cid:17)
j−1 j D (v)(cid:44)vM − .
where we have used the fact that vv ≥ vv in the coupling g uM 2 uM
j j−1
set Cs. Claim(ii) now follows bysubstituting uvj =gsf =gus Proof: First notice from (14) that uvi,j(t)≤guc, for all t
and using the definition of gus. A similar argument can be such that ∃Fi,j. Further notice that if at some time t1, (cid:64)Fi,j
used to show claim (iii). then it remains (cid:64)F for all t ≥ t for otherwise it means
i,j 1
(iv) Setting vjv =vM in the definition of gus and using the there exists some control policy starting from t=t1 such that
fact that gus ≥0, we have Tia,j =τi,j andviv,j(Tia,j)≥νnom andRemarkVII.5guarantees
vv ≥ −umvM . ∃FNi,ojwa,t tth=eret1.arTehutws,ofocraseeasch-veethhiecrleth(ie,jo)p,tTimia,jal≥coτin,jtr.oller
j−1 −u +σ uv
m j j−1 exists until the vehicle reaches the intersection or it becomes
Toobtainthenecessaryconditiononvv ,wesetuv =u infeasible earlier. We consider each of these cases separately.
j−1 j−1 M
and σ =σ , the maximum values for each. In the first case, notice that for any vehicle (i,j), for j ∈
j 0
InLemmaVII.6,claim(i)statesthatg respectsthecontrol {2,...,m }, if ∃F at t = Ta , then it follows from the
us i i,j i,j
constraints.Claims(ii)and(iii)givesomesufficientconditions definition of Ta that Ta = τ and vv (Ta ) ≥ νnom and
i,j i,j i,j i,j i,j
for ensuring the safety ratio σ is constant. Claim (iii) also from (10), we have that Ta −Ta ≤Tnom.
j i,j i,j−1