Robust Distributed Control Protocols for Large Vehicular Platoons with Prescribed Transient and Steady State Performance

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by  Christos K. Verginis

1
Robust Distributed Control Protocols for Large
Vehicular Platoons with Prescribed Transient and
Steady State Performance
Christos K. Verginis, Charalampos P. Bechlioulis, Dimos V. Dimarogonas and Kostas J. Kyriakopoulos
Abstract—In this paper, we study the longitudinal control [12]acombinedpredecessorandleader-followingarchitecture
problem for a platoon of vehicles with unknown nonlinear was developed according to which each vehicle obtains addi-
7
dynamics under both the predecessor-following and the bidirec-
1 tionalinformationfromtheleadingvehicle.Finally,[13],[14]
tional control architectures. The proposed control protocols are
0 addressed various architectures by examining different kinds
fully distributed in the sense that each vehicle utilizes feedback
2 from its relative position with respect to its preceding and of information flow topologies.

n following vehicles as well as its own velocity, which can all Themajorityoftheworksintherelatedliteratureeithercon-
a be easily obtained by onboard sensors. Moreover, no previous siderslinearvehicledynamicmodelsandcontrollers[8],[15],
J knowledge of model nonlinearities/disturbances is incorporated
[16] or adopt linearization techniques and Linear Quadratic
3 in the control design, enhancing in that way the robustness
optimalcontrol[6],[9],[13],[14].However,linearizationmay
2 of the overall closed loop system against model imperfections.
Additionally, certain designer-specified performance functions lead to unstable inner dynamics since the estimated linear
] determine the transient and steady-state response, thus prevent- models deviate in general from the real ones, away from the
Y ingconnectivitybreaksduetosensorlimitationsaswellasinter- correspondinglinearizationpoints.Inparticular,acomparison
vehicular collisions. Finally, extensive simulation studies and a
S of the aforementioned control architectures was carried out in
real-time experiment conducted with mobile robots clarify the
. [16], where it was stated that double integrator models with
s proposed control protocols and verify their effectiveness.
c linear controllers under the predecessor-following architecture
[
mayleadtostringinstability.Stringinstabilityconditionswere

1 I. INTRODUCTION also presented in [17]. Finally, in [18] a comparison of two
v common control policies was conducted; namely the constant
DURING the last few decades, Automated Highway Sys-
8 time headway policy and the constant spacing policy, that
3 tems(AHS)havedrawnanotableamountofattentionin
are related to the inter-vehicular distances of the platoon.
4 the field of automatic control. Unlike human drivers, that are
Particularlyforthelatter,itwasalsostatedthatfeedbackfrom
6 notabletoreactquicklyandaccuratelyenoughtofolloweach
0 the leading vehicle needs to be constantly broadcasted.
otherincloseproximityathighspeeds,thesafetyandcapacity
. Another important issue associated with the decentralized
1 of highways (measured in vehicles/lanes/time) is significantly
0 increased when vehicles operate autonomously forming large control of large platoons of vehicles, concerns the fact that
7 in many works the transient and steady state response of
platoons at close spacing.
1 the closed loop system is affected severely by the control
: Guaranteed string stability [1] was first achieved via cen-
v gains’ selection and the number of vehicles as stated in
tralized control schemes [2]–[4], with all vehicles either com-
i [10], [11], [16], limiting thus the controller’s capabilities.
X municating explicitly with each other or sending information
Furthermore,themajorityoftheresultsontheaforementioned
r to a central computer that determined the control protocol.
a To enhance the overall system’s autonomy and avoid delay decentralized architectures consider known (either partially of
fully) dynamic models and parameters, which may lead to
problems due to wireless communication [5], decentralized
poor closed loop performance in the presence of parametric
schemes were developed, dealing either with the predecessor-
uncertainties and unknown external disturbances.
following (PF) architecture [6]–[8], where each vehicle has
Inthiswork,weproposedecentralizedcontrolprotocolsfor
access to its relative position with respect to its preceding
large platoons of vehicles with 2nd order1 uncertain nonlinear
vehicle, or the bidirectional (BD) architecture [9]–[11], where
dynamics, under both the predecessor-following and the bidi-
each vehicle measures its relative position with respect to its
rectional control architectures. The desired feasible formation
following vehicle as well. Furthermore, in a few works [5],
iscreatedarbitrarilyfastandismaintainedwitharbitraryaccu-
racy avoiding simultaneously any connectivity breaks (owing
C.K.VerginisandD.V.DimarogonasarewiththeCentreforAutonomous
Systems at Kungliga Tekniska Hogskolan, Stockholm 10044, Sweden. C. P. to limited sensor capabilities) and inter-vehicular collisions.
BechlioulisandK.J.KyriakopoulosarewiththeControlSystemsLaboratory, The developed schemes exhibit the following significant char-
SchoolofMechanicalEngineering,NationalTechnicalUniversityofAthens,
Athens 15780, Greece. Emails: [email protected], [email protected],
[email protected],[email protected]. 1The results may be easily extended for 3rd order dynamics, that model
This work was supported by the EU funded project RECONFIG: Cog- thedriving/brakingforcewithafirst-orderinertialtransferfunction[13],[14],
nitive, Decentralized Coordination of Heterogeneous Multi-Robot Systems followingsimilardesignstepsaspresentedin[19].However,inthiswork,we
viaReconfigurableTaskPlanning(FP7-ICT-600825,2013-2016),theSwedish adopted a 2nd order model only to present more clearly the control design
ResearchCouncil(VR)andtheKnutandAliceWallenbergfoundation. philosophyandhighlightitsproperties.
2
acteristics. First, they are purely distributed in the sense that v (t). Finally, to solve the aforementioned formation control
0
the control signal of each vehicle is calculated based solely: problem, the following assumption is required.
a) on local relative position information with respect to its Assumption A1. The initial state of the platoon does
preceding and following vehicles, as well as b) on its own not violate the collision and connectivity constraints. That is
velocity, both of which can be easily acquired by its on-board ∆ < p (0)−p (0)<∆ , i=1,...,N.
col i−1 i con
sensors. Furthermore, their complexity proves to be consid- In this work, we consider two distributed control archi-
erably low. Very few and simple calculations are required to tectures: (a) the predecessor-following (PF) architecture, ac-
output the control signals. Additionally, they do not require cording to which the control action of each vehicle is based
any previous knowledge of the vehicle’s dynamic model only on its preceding vehicle and (b) the bidirectional (BD)
parameters and no estimation models are employed to acquire architecture, where the control action of each vehicle depends
such knowledge. Moreover, contrary to the related works, the on the information from both its preceding and following
transientandsteadystateresponseisfullydecoupledby:i)the vehicles.Hence,letusformulatethecontrolvariablese (t)=
pi
numberofvehiclescomposingtheplatoon,ii)thecontrolgains p (t) − p (t) − ∆ , i = 1,...,N. Equivalently, the
i−1 i i−1,i
selectionandiii)thevehiclemodeluncertainties.Inparticular, neighborhood error vector e (cid:44) [e ,...,e ]T may be
p p1 pN
the achieved performance as well as the collision avoidance expressed as follows:
and the connectivity maintenance are a priori and explicitly
e =Se (2)
imposed by certain designer-specified performance functions, p p0
thussimplifyingsignificantlytheselectionofthecontrolgains. wheree (cid:44)[e ,...,e ]T =p¯ −p−∆¯ istheerrorwith
p0 p0,1 p0,N 0 0
Tuning of the controller gains is only confined to achieving respect to the leading vehicle, p(cid:44)[p ,...,p ]T ∈(cid:60)N, p¯ (cid:44)
1 N 0
reasonable control effort. [p ,...,p ]T ∈ (cid:60)N, ∆¯ (cid:44) [∆ ,∆ ,...,∆ ]T ∈ (cid:60)N
0 0 0 0,1 0,2 0,N
with ∆ = (cid:80)i ∆ , i = 1,...,N and S ∈ (cid:60)N×N =
0,i j=1 j−1,j
[s ] where s = 1, s = −1 and s = 0 for all other
II. PROBLEMSTATEMENT i,j i,i i+1,i i,j
elements, with i,j = 1,...,N. Notice that S has strictly
We consider the longitudinal formation control problem of positivesingularvalues[3]andsinceallprincipalminorsofS
N vehicles with 2nd order nonlinear dynamics: areequalto1,SisalsoanonsingularM-matrix2[20].Finally,
the following lemma regarding nonsingular M-matrices will
p˙ =v
i i be employed to derive the main results of this paper.
m v˙ =f (v )+u +w (t), i=1,...,N (1)
i i i i i i
Lemma 1. [3] Consider a nonsingular M-matrix A ∈
where p and v denote the position and velocity of each (cid:60)N×N. There exists a diagonal positive definite matrix P =
i i
vehicle respectively, m is the mass, which is considered (diag(A−11))−1 such that PA+ATP is positive definite.
i
unknown,f (v )isanunknowncontinuousnonlinearfunction
i i
that models the aerodynamic friction (drag), ui is the control III. MAINRESULTS
input and w (t) is a bounded piecewise continuous function
i In this work, prescribed performance will be adopted in or-
of time representing exogenous disturbances. The control
der:i)toachievepredefinedtransientandsteadystateresponse
objective is to design a distributed control protocol such
for each neighborhood position error e (t), i = 1,...,N
that a rigid formation is established with prescribed transient pi
as well as ii) to avoid the violation of the collision and
and steady state performance, despite the presence of model
connectivity constraints as presented in Section II.
uncertainties. By prescribed performance, we mean that the
formation is achieved in a predefined transient period and
A. Sufficient Conditions
is maintained arbitrarily accurate while avoiding connectivity
breaksandcollisionswithneighboringvehicles.Thegeometry Prescribedperformanceisachievedwhentheneighborhood
of the formation is represented by the desired gaps ∆ , position errors e (t), i = 1,...,N evolve strictly within
i−1,i pi
i=1,...,N betweenconsecutivevehicles,where∆ >0 predefined regions that are bounded by absolutely decaying
i−1,i
denotes the desired distance between the (i − 1)-th and i- functions of time, called performance functions [19], [21]. In
th vehicle. In general, all ∆ are given as control spec- this work, the mathematical expression of prescribed perfor-
i−1,i
ifications and are directly related to the platoon velocity as mance is formulated by the following inequalities:
well as to the input constraints of the traction/bracking forces
−M ρ (t)<e (t)<M ρ (t), ∀t≥0 (3)
(for safety reasons). Moreover, the inter-vehicular distance pi pi pi pi pi
pi−1(t) − pi(t), i = 1,...,N should be kept greater than for all i=1,...,N, where:
∆ to avoid collisions and less than ∆ to maintain the
col con (cid:18) (cid:19)
network connectivity owing to the limited sensing capabilities ρ (t)= 1− ρ∞ exp(−lt)+ ρ∞
ofthevehicles(e.g.,whenemployingrangesensorstomeasure pi max{Mpi,Mpi} max{Mpi,Mpi}
(4)
the distance between two successive vehicles). Furthermore,
are designer-specified, smooth, bounded and decreasing func-
to ensure the feasibility of the desired formation, we assume
tions of time with l, ρ positive parameters incorporating the
that ∆ < ∆ < ∆ , i = 1,...,N. Additionally, ∞
col i−1,i con
the reference command of the formation is generated by
2An M-matrix is a square matrix whose off-diagonal elements are less
a leading vehicle with position p0(t) and bounded velocity thanorequaltozeroandwhoseeigenvalueshavepositiverealpart.
3
desired transient and steady state performance specifications A. Predecessor-Following architecture:
rseeslepcetcetdivaeplyp,raonpdriaMteplyi,tMo spait,isify=t1h,e.c.o.l,lNisiopnosaintidvecopnanraemcteivteitrys  vd1(ξp1,t) 
.
 . 
constraints, as presented in the sequel. In particular, the v (ξ ,t)(cid:44) . 
d p  
decreasing rate of ρpi(t), i = 1,...,N, which is affected vdN−1(ξpN−1,t)
by the constant l, introduces a lower bound on the speed of v (ξ ,t)
dN pN
convergence of epi(t), i=1,...,N. Furthermore, depending =kp(ρp(t))−1rp(ξp)εp(ξp), (10)
on the accuracy of the measurement device, the constant ρ
∞
can be set arbitrarily small ρ (cid:28) M ,M , i = 1,...,N, B. Bidirectional architecture:
thus achieving practical conve∞rgence opfiepi(pti), i = 1,...,N  vd1(ξp1,ξp2,t) 
to zero. Additionally, we select: .
 . 
v (ξ ,t)(cid:44) . 
Mpi =∆i−1,i−∆col & Mpi =∆con−∆i−1,i, i=1,...,(N5). d p vdN−v1(ξp(Nξ−1,,ξtp)N,t)
Apparently,sincethedesiredformationiscompatiblewiththe dN pN
=k ST(ρ (t))−1r (ξ )ε (ξ ) (11)
collision and connectivity constraints (i.e., ∆ < ∆ < p p p p p p
col i−1,i
∆con, i = 1,...,N), (5) ensures that Mpi, Mpi > 0, with kp >0.
i = 1,...,N and consequently under Assumption A1 (i.e., Dynamic Controller:
∆col < pi−1(0)−pi(0)<∆con, i=1,...,N) that: Step II-a. Define the velocity error vector ev (cid:44)
[e ,...,e ]T = v − v (ξ ,t) with v (cid:44) [v ,...,v ]T
−Mpiρpi(0)<epi(0)<Mpiρpi(0), i=1,...,N. (6) fovr1both cvoNntrol architectudrespand select the co1rrespondNing
Hence, guaranteeing prescribed performance via (3) for all velocity performance functions ρ (t), i = 1,...,N such
vi
t > 0 and employing the decreasing property of ρpi(t), that ρvi(0)>|evi(0)|, i=1,...,N.
i=1,...,N, we obtain −Mpi <epi(t)<Mpi, ∀t≥0 and Step II-b. Similarly to the first step define the normalized
consequently,owingto(5),∆col <pi−1(t)−pi(t)<∆con for velocity errors as:
allt≥0andi=1,...,N,whichensurescollisionavoidance
and connectivity maintenance for all t≥0. Therefore, impos- ξv1(ev1,t) ρve1v(1t)
ipcnoegrnfspotrraemnstacnrpicabereadmfupenetrceftroisormnMsapnρic,peiMv(tipa)i,,(3ii)=w=i1th,1.,a..p..p,.rN,oNp,riaasatendldyicstpaeotleseidcttievidne ξv(ev,t)(cid:44)ξvN(e...vN,t)(cid:44)ρveNv...N(t)≡(ρv(t))−1ev(,12)
(4) and (5) respectively, proves sufficient to solve the robust
where ρ (t) = diag([ρ (t)] ) as well as the control
formation control problem stated in Section II. v vi i=1,...,N
signals:
B. Control Design (cid:18)(cid:104) (cid:105) (cid:19)
r (ξ )=diag 2 (13)
Kinematic Controller: Given the neighborhood position er- v v (1+ξvi)(1−ξvi) i=1,...,N
rorSsteeppiI(-ta).=Spelie−c1t(tt)h−e pcoi(rtr)es−po∆ndi−in1g,i,fiu=nct1io,.n.s.,ρNpi:(t) and εv(ξv)=(cid:104)ln(cid:16)11−+ξξvv11(cid:17),...,ln(cid:16)11−+ξξvvNN(cid:17)(cid:105)T . (14)
positiveparametersMpi,Mpi,i=1,...,N following(4)and Step II-c. Design the distributed control protocol for both
(5) respectively, in order to incorporate the desired transient architectures as follows:
and steady state performance specifications as well as the  
u (ξ ,t)
collision and connectivity constraints. 1 v1
Step I-b. Define the normalized position errors as: u(ξv,t)(cid:44) ... =−kv(ρv(t))−1rv(ξv)εv(ξv)
ξ (e ,t)  ep1  uN(ξvN,t)
p1 p1 ρp1(t) (15)
ξp(ep,t)(cid:44) ... (cid:44) ... ≡(ρp(t))−1ep, (7) with kv >0.
ξpN(epN,t) ρpeNpN(t) Remark1. (ControlPhilosophy)Theprescribedperformance
control technique guarantees the prescribed transient and
where ρ (t) = diag([ρ (t)] ) as well as the control
p pi i=1,...,N steady state performance specifications, that are encapsulated
signals:
in (3), by enforcing the normalized position errors ξ (t)
(cid:34) (cid:35)  pi
εrpp((ξξpp))==d(cid:34)ilang(cid:32)11−+(Mξξ1pp+p111MMξ(cid:33)p1ppiii,)+.(1.M−.1p,Miξlnppii(cid:32))11−+i=Mξξ1pp,p.NNN..,N(cid:33)(cid:35)T . ((89)) taawhlnieltdhtliovnge≥latohrciet0iht.ysmeNeitcsrortfo(cid:0)iucr−nescMtξtihvopianit(,stM)ml,nopi(cid:18)id(cid:1)u=11l−+aatnMM1idn(cid:63),(cid:63)ppg.ii.((cid:19)−.ξ,p1Nai,n(1td))tolrnaerne(cid:16)sdpm11e−+acξi(cid:63)(cid:63)tvnii(cid:17)v(estlit)ynrictvfthoilayer
Mp1 MpN control signals (9), (14) and selecting Mpi, Mpi according
Step I-c. Design the reference velocity vector for the to (5) and ρ (0) > |e (0)|, the control signals ε (ξ ) and
vi vi p p
predecessor-following and bidirectional control architectures ε (ξ ) are initially well defined. Moreover, it is not difficult
v v
as follows: to verify that maintaining simply the boundedness of the
4
modulated errors ε (ξ (t)) and ε (ξ (t)) for all t ≥ 0 we prove that the proposed control scheme retains ξ (t)
p p v v (cid:0) (cid:1) (cid:0) (cid:1)pi
is equivalent to guaranteeing ξ (t) ∈ −M ,M and and ξ (t) strictly in compact subsets of −M ,M and
pi pi pi vi pi pi
ξ (t) ∈ (−1,1) for all t ≥ 0. Therefore, the problem at (−1,1) for all t ∈ [0,τ ), which by contradiction leads to
vi max
hand can be visualized as stabilizing the modulated errors τ =∞ (i.e., forward completeness) in the last phase, thus
max
ε (ξ (t)) and ε (ξ (t)), within the feasible regions defined completingtheproof.Itshouldbealsonoticedthattheproofis
p p v v
(cid:0) (cid:1)
via ξ ∈ −M ,M and ξ ∈ (−1,1), i = 1,...,N for provided only for the predecessor-following architecture since
pi pi pi vi
all t≥0. A careful inspection of the proposed control scheme the proof for the bidirectional case follows identical steps.
(10),(11)and(15)revealsthatitactuallyoperatessimilarlyto In particular, by differentiating (7) and (12), we obtain:
barrier functions in constrained optimization, admitting high
ξ˙ =(ρ (t))−1(e˙ −ρ˙ (t)ξ ) (16)
negative or positive values depending on whether e (t) → p p p p p
−Mpiρpi(t) and evi(t) → ρvi(t) or epi(t) → Mppiiρpi(t) ξ˙v =(ρv(t))−1(e˙v−ρ˙v(t)ξv) (17)
and e (t)→−ρ (t), i=1,...,N respectively; eventually
prevenvtiing e (t)vaind e (t), i=1,...,N from reaching the Employing(1),(2)aswellasthefactthatvi ≡vdi+ρvi(t)ξvi
pi vi and substituting (10), (15) in (16) and (17), we arrive at:
corresponding boundaries.
ξ˙ =h (t,ξ)
Remark 2. (Selecting the Performance Functions) Regard- p pA
=−k (ρ (t))−1S(ρ (t))−1r (ξ )ε (ξ )
ing the construction of the performance functions, we stress p p p p p p p
that the desired performance specifications concerning the −(ρ (t))−1(ρ˙ (t)ξ +S(ρ (t)ξ −p˙ (t))) (18)
p p p v v 0
transient and steady state response as well as the collision ξ˙ =h (t,ξ)
and connectivity constraints are introduced in the proposed v vA
=−k (ρ (t))−1M−1ε (ξ )−(ρ (t))−1(ρ˙ (t)ξ
control schemes via ρ (t) and M , M , i = 1,...,N v v v v v v v
respectively. In additiopni, the velocitypiperfoprimance functions −M−1(f(vd+ρv(t)ξv)+w(t))+v˙d) (19)
ρ (t) impose prescribed performance on the velocity errors (cid:16) (cid:17)
vi where M = diag [m ] and f(v + ρ (t)ξ ) =
eavcit =asvrie−fervednic,eis=ign1a,l.s..fo,Nr t.hIenctohrisrersepsopnedcitn,gnovteicloectihtiaetsvvdi, [f (v +ρ (t)ξ ),··i·i,=f1,..(.,vN +ρ (t)dξ ]Tv withv m ,
i 1 d1 v1 v1 N dN vN vN i
i = 1,...,N. However, it should be stressed that although f (·), i=1,...,N denoting the unknown masses and nonlin-
i
such performance specifications are not required (only the earities ofthe vehiclemodel (1)respectively. Thus, theclosed
neighborhood position errors need to satisfy predefined tran- loop dynamical system of ξ(t) = (cid:2)ξT(t),ξT(t)(cid:3)T may be
p v
sient and steady state performance specifications) their selec- written in compact form as:
tionaffectsboththeevolutionofthepositionerrorswithinthe (cid:20) (cid:21)
h (t,ξ)
corresponding performance envelopes as well as the control ξ˙=h (t,ξ)(cid:44) pA . (20)
A h (t,ξ)
input characteristics (magnitude and rate). Nevertheless, the vA
only hard constraint attached to their definition is related to Let us also define the open set Ω =Ω ×Ω ⊂(cid:60)2N with:
ξ ξp ξv
their initial values. Specifically, ρ (0) should be chosen to
vi Ω =(−M ,M )×···×(−M ,M )
satisfy ρvi(0)>|evi(0)|, i=1,...,N. ξp p1 p1 pN pN
Ω =(−1,1)×···×(−1,1). (21)
ξv
(cid:124) (cid:123)(cid:122) (cid:125)
C. Stability Analysis N-times
Themainresultsofthisworkaresummarizedinthefollow- Phase I. Selecting the parameters M , M , i=1,...,N
pi pi
ing theorem, where it is proven that the aforementioned dis- according to (5), we guarantee that the set Ω is nonempty
ξ
tributed control protocols solve the robust formation problem and open. Moreover, owing to Assumption A1, ξ (0)∈Ω ,
p ξp
with prescribed performance under collision and connectivity as shown in (6). Furthermore, selecting ρ (0) > |e (0)|,
vi vi
constraints for the considered platoon of vehicles. i = 1,...,N ensures that ξ (0) ∈ Ω as well. Thus, we
v ξv
conclude that ξ(0) ∈ Ω . Additionally, h is continuous on
Theorem 1. Consider a platoon of N vehicles with uncertain ξ A
t and locally Lipschitz on ξ over the set Ω . Therefore, the
2nd order nonlinear dynamics (1), that aims at establishing ξ
hypotheses of Theorem 54 in [22] (p.p. 476) hold and the
a formation described by the desired inter-vehicular gaps
existenceofamaximalsolutionξ(t)of(20)foratimeinterval
∆ , i = 1,...,N, while satisfying the collision and con-
i−1,i [0,τ ) such that ξ(t)∈Ω , ∀t∈[0,τ ) is guaranteed.
nectivityconstraintsrepresentedby∆ and∆ respectively max ξ max
col con PhaseII-Kinematics.WehaveproveninPhaseI thatξ(t)∈
with ∆ < ∆ < ∆ , i = 1,...,N. Under Assump-
col i−1,i con Ω , ∀t∈[0,τ ) and more specifically that:
tion A1, the distributed control protocols (7)-(15), for the ξ max

pgtru≥eadrea0cneatsensedo:r-i−fo=Mllo1pwi,ρi.np.gi.,(atNn)d,<absidewipreeilc(ltti)aosn<atlhecMobnpotiruρonpldiae(rdtc)nhe,istesfcootrufreaasllll, ξξvpii((tt))== ρρeevpvpiiii((((tttt)))) ∈∈((−−1M,1p)i, Mpi)  i=1,...,N (22)
closed loop signals.
for all t ∈ [0,τ ), from which we obtain that e (t) and
max pi
Proof: The proof of Theorem 1 proceeds in three phases. e (t) are absolutely bounded by max{M , M }ρ (t) and
vi pi pi pi
First, we show that ξ (t) and ξ (t) remain within ρ (t) respectively for i = 1,...,N. Furthermore, owing
(cid:0) (cid:1) pi vi vi
−M ,M and (−1,1) respectively, for a specific time to (22), the error vector ε (t), as defined in (9), is well
pi pi p
interval[0,τ )(i.e.,existenceofamaximalsolution).Next, defined for all t ∈ [0,τ ). Therefore, consider the positive
max max
5
definite and radially unbounded function V = 1εTPε , w(t). Furthermore, from (14), taking the inverse logarithmic
pA 2 p p
where P (cid:44) (diag(S−11))−1 is a diagonal positive definite function, we obtain:
matrix satisfying PS +STP > 0, as dictated by Lemma 1.
−1<−exp(ε¯v)−1=ξ ≤ξ (t)≤ξ¯ =exp(ε¯v)−1<1 (28)
Differentiating VpA with respect to time, substituting (8), (18) exp(ε¯v)+1 vi vi vi exp(ε¯v)+1
and exploiting: i) the diagonality of the matrices P, r (ξ ),
p p for all t∈[0,τ ) and i=1,...,N, which also leads to the
ρ (t), ii) the positive definiteness of Q(cid:44)PS+STP as well max
p boundedness of the distributed control protocol (15).
as iii) the boundedness of ρ˙ (t), ρ (t) and p˙ (t), we get:
p v 0 PhaseIII.Uptothispoint,whatremainstobeshownisthat
V˙pA +≤(cid:13)(cid:13)−εkTpprλpm(ξinp()Q(ρ)p(cid:13)(cid:13)(εt)Tp)r−p1((cid:13)(cid:13)ξpF¯)p(ρp(t))−1(cid:13)(cid:13)2 τaΩmn(cid:48)dax((cid:44)2c8a(cid:104))nξtbhea,tξ¯eξx(t(cid:105)et)n×d∈e·d·Ω·t(cid:48)ξ×o(cid:44)∞(cid:104)ξΩ.(cid:48)ξIpn,ξׯthΩis(cid:105)(cid:48)ξdvai,nre∀dcttΩi∈o(cid:48)n[,0(cid:44),nτo(cid:104)mtiξacxe),,bξ¯ywh((cid:105)2e5×re)
ξp (cid:104) p1 p1(cid:105) pN pN ξv v1 v1
where Fp is a positive constant independent of τmax, satisfy- ···× ξ ,ξ¯ are nonempty and compact subsets of Ω
ing: vN vN ξp
and Ω respectively. Hence, assuming τ < ∞ and since
(cid:13)(cid:13)P(ρ˙p(t)ξp+S(ρv(t)ξv−p˙0(t)))(cid:13)(cid:13)≤F¯p (23) Ω(cid:48)ξ ⊂ξΩv ξ, Proposition C.3.6 in [22] (p.mp.ax481) dictates the
existenceofatimeinstantt(cid:48) ∈[0,τ )suchthatξ(t(cid:48))∈/ Ω(cid:48),
for all (ξ,t) ∈ Ω ×(cid:60) . Therefore, we conclude that V˙ max ξ
is negative whenξ(cid:13)(cid:13)εTprp+(ξp)(ρp(t))−1(cid:13)(cid:13) > kpλmF¯ipn(Q), fropmA w∞h.iTchhuiss,aalcllcelaorsecdonltoroapdiscitgionna.lsTrheemreafinorbeo,uτnmdaexdiasnedxmtenodreeodvteor
which, owing to the positive definiteness and diagonality of ξ(t) ∈ Ω(cid:48) ⊂ Ω , ∀t ≥ 0. Finally, multiplying (25) by ρ (t),
r (ξ )(ρ (t))−1 as well as employing (8) and (4), it can be ξ ξ pi
p p p i=1,...,N, we also conclude that:
easily verified that:
−M ρ (t)<e (t)<M ρ (t), ∀t≥0 (29)
(cid:107)ε (t)(cid:107)≤ε¯ := λmax(P)max(cid:107)ε (0)(cid:107),F¯pmax(cid:40)MMppii+MMppii(cid:41)for all i =pi1,p.i..,N apnid consequpeinptliy the solution of the
p p λmin(P)  p kpλmin(Q) robustformationcontrolproblemwithprescribedperformance
undercollisionandconnectivityconstraintsfortheconsidered
(24)
platoon of vehicles. (cid:4)
for all t∈[0,τ ). Furthermore, from (9), taking the inverse
max
logarithm, we obtain: Remark 3. From the aforementioned proof it can be deduced
that the proposed control schemes achieve their goals without
−M <− exp(ε¯p)−1 M =ξ ≤ξ (t) resorting to the need of rendering the ultimate bounds ε¯ ,
pi exp(ε¯p)+MMppii pi pi pi ε¯v of the modulated position and velocity errors εp(ξp(t)p),
≤ξ¯pi=exepx(pε¯(pε¯)p+)−MM1ppii Mpi<Mpi (25) εthve(ξcvo(ntt)r)olargbaitirnasrilkyp samnadll kbvy (asdeoept(i2n4g)eaxtnrdem(e26v)a).lueMsoroef
specifically, notice that (25) and (28) hold no matter how
for all t ∈ [0,τ ) and i = 1,...,N. Thus, the reference
max large the finite bounds ε¯ , ε¯ are. In the same spirit, large
p v
velocityvectorv (ξ ,t),asdesignedin(10),remainsbounded
d p uncertainties involved in the vehicle nonlinear model (1) can
for all t∈[0,τmax). Moreover, invoking vi ≡vdi+ρvi(t)ξvi be compensated, as they affect only the size of ε¯v through F¯v
we also conclude the boundedness of the velocities v (t), i=
i (see(27)),butleaveunalteredtheachievedstabilityproperties.
1,...,N for all t∈[0,τ ). Finally, differentiating v (ξ ,t)
max d p Hence, the actual performance given in (29), which is solely
with respect to time, substituting (18) and utilizing (25), it
determined by the designer-specified performance functions
is straightforward to deduce the boundedness of v˙ for all
d ρ (t) and the parameters −M , M , i = 1,...,N,
t∈[0,τ ) as well. pi pi pi
max becomes isolated against model uncertainties, thus extending
Phase II-Dynamics. Owing to (22), the error vector ε (t)
v greatly the robustness of the proposed control schemes.
(see (14)) is well defined for all t ∈ [0,τ ). Therefore,
max
consider the positive definite and radially unbounded function Remark 4. (Selecting the Control Gains) It should be noted
(cid:16) (cid:17)
V = 1εTMε where M = diag [m ] with m , that the selection of the control gains affects both the quality
vA 2 v v i i=1,...,N i of evolution of the neighborhood errors e (t), i=1,...,N
i=1,...,N denotingtheunknownmassofthevehiclemodel pi
inside the corresponding performance envelopes as well as
(1). Following the same line of proof with V in Phase II-
pA the control input characteristics (e.g., decreasing the gain
Kinematics, we conclude that:
values leads to increased oscillatory behaviour within the
(cid:110) (cid:111)
(cid:107)εv(t)(cid:107)≤ε¯v := mmainx{{mmii}}max (cid:107)εv(0)(cid:107),2F¯kvv (26) pimrepsrcorviebdedwpheernforamdoapntciengenhvieglhoeprevdaelsucersib,eednlbayrg(i2n9g),,hwohwicehveirs,
forallt∈[0,τ ),whereF¯ isapositiveconstantsatisfying: the control effort both in magnitude and rate). Additionally,
max v
finetuningmightbeneededinreal-timescenarios,toretainthe
(cid:107)(Mρ˙ (t)ξ −(f(v +ρ (t)ξ )+w(t))+v˙ )(cid:107)≤F¯ required control input signals within the feasible range that
v v d v v d v
(27) can be implemented by the actuators. Similarly, the control
owing to: i) the boundedness of v and v˙ that was proven input constraints impose an upper bound on the required
d d
previously, ii) the continuity of function f (·) and iii) the speed of convergence of ρ (t), i = 1,...,N, as obtained
i pi
boundednessofρ˙ (t),ρ (t)aswellasofthedisturbanceterm by the exponentials exp(−lt). Hence, the selection of the
v v
6
etm()[]p1−055 −00..011 etm()[]p2−055 −00..011 utkNt()[]1−1310500 utkNt()[]2−1310500
etm()[]p3−055 −00..011 etm()[]p4−055 −00..011 utkNt()[]3−1310500 utkNt()[]4−1315000
etm()[]p5−055 −00..011 etm()[]p6−055 −00..011 utkNt()[]5−1310500 utkNt()[]6−1315000
etm()[]p7−055 −00..011 etm()[]p8−055 −00..011 utkNt()[]7−1310500 utkNt()[]8−1310500
etm()[]p9−0550 −2000..011 40t[sec]60 80 100 12etm0()[]p10−0550 2−000..011 40t[sec]60 80 100 120 utkNt()[]9−13105000 20 40t[sec]60 80 100 12utkNt0()[]10−13105000 20 40t[sec]60 80 100 120
Fig.1. Thepositionerrorsepi(t)(PF). Fig.2. Therequiredcontrolinputsignals(PF).
control gains k and k can have positive influence on the aforementionedparametersdoesnotaffecttheperformanceof
p v
overall closed loop system response. More specifically, notice theproposedschemes.Furthermore,theleadingvehicleadopts
that (23)-(28) provide bounds on ε and ε that depend on the following continuous velocity profile:
p v
the constants Fp and Fv. Therefore, invoking (10),(11) and  75t2−t3, t∈[0,50]
(a1n5d)uwearceanrestaeliencetdthweithcoinntcroelrtgaaininbsokupndasn.dNkevvesruthcehletshsa,ttvhde  252,500 t∈[50,70]
v (t)= 0.02t3−4.5t2+336t−8305, t∈[70,80]
constantsF andF involvetheparametersofthemodel,the 0
external distpurbancevs, the velocity/acceleration of the leader  1157,.5−2.5cos(cid:0)t−90(cid:1), tt∈∈[9[800,1,9200]]
and the desired performance specifications. Thus, an upper 2
bound of the dynamic parameters of the system as well as whereas the desired distance between consecutive vehicles is
of the exogenous disturbances must be given in order to equally set at ∆ = ∆(cid:63) = 4 m, i = 1,...,10 with the
i−1,i
extract any relationships between the achieved performance collisionandconnectivityconstraintsgivenby∆ =0.05∆(cid:63)
col
and the input constraints3. Finally, in the same direction, and ∆ = 1.95∆(cid:63) respectively. Notice that the aforemen-
con
the selection of the velocity performance functions ρvi(t), tionedformationproblemunderthecollision/connectivitycon-
i,...,N affects both the evolution of the position errors straintsisfeasiblesince∆ <∆ <∆ ,i=1,...,10.
col i−1,i con
within the corresponding performance envelopes as well as Moreover, we require steady state errors of no more than
the control input characteristics. 0.05 m and minimum speed of convergence as obtained by
the exponential exp(−0.1t). Thus, according to (4) and (5),
Remark5. (StringStability)Notethattheproposedalgorithm
we selected the parameters M = M = 0.95∆(cid:63) and
guarantees string stability for the equilibrium point epi = the functions ρ (t) = (1 − 0p.i05 )expp(i−0.1t) + 0.05 ,
0,i=1,...,N, in the sense of [1] (see Def. 1). In particular, pi 0.95∆(cid:63) 0.95∆(cid:63)
i = 1,...,10 in order to achieve the desired transient and
forany(cid:15)>0,wecanchooseδ =max {max{M ,M }}=
i pi pi steady state performance specifications as well as to comply
(cid:15), i = 1,...,N. Then, from the aforementioned anal-
with the collision and connectivity constraints. Moreover, we
ysis it can be deduced that max |e (0)| < δ implies
i pi chose ρ (t)=2|e (0)|exp(−0.1t)+0.1 in order to satisfy
maxi{supt≥0|epi(t)|}<(cid:15), i=1,...,N. ρ (0) v>i |e (0)|v,ii = 1,...,10. Finally, in view of the
vi vi
desired motion profile of the leader as well as the masses
IV. SIMULATIONRESULTS
of the vehicles, we chose the control gains as k = 0.1,
p
A. Generic Evaluation k = 100 for the predecessor-following architecture and
v
To demonstrate the efficiency of the proposed distributed k = 10, k = 1000 for the bidirectional architecture, to
p v
controlprotocols,weconsideredaplatoonofN =10vehicles obtain control inputs that satisfy |u |≤30 kNt, i=1,...,10.
i
obeying (1) with f (v ) = −50v − 25|v |v , w (t) = ThesimulationresultsareillustratedinFigs.1,2and3,4for
i i i i i i
A sin(ω t+φ ) and m , A , ω , φ randomly selected within thepredecessor-following(PF)andthebidirectional(BD)con-
i i i i i i i
[500,1500] kg, [1.0, 1.5] kNt, [2π, 4π] rad/s and [0, 2π] rad trol architectures respectively. More specifically, the evolution
respectively for i = 1,...,10. Although the size of the oftheneighborhoodpositionerrorse (t),i=1,...,10along
pi
aforementioned intervals affects directly the magnitude of the with the corresponding performance functions are depicted in
control effort u, which however can be regulated by tuning Figs. 1 and 3, while the required control inputs are illustrated
appropriately the gains k and k , as mentioned in Remark inFigs.2and4.Asitwaspredictedbythetheoreticalanalysis,
p v
4, in view of the theoretical analysis, the uncertainty of the the formation control problem with prescribed transient and
steady state performance is solved with bounded closed loop
3Noticethattheproposedmethodologydoesnottakeexplicitlyintoaccount signals for both control architectures, despite the presence of
anyspecificationsintheinput(magnitudeorslewrate).Suchresearchdirec-
external disturbances as well as the lack of knowledge of the
tionisanopenissueforfutureinvestigationandwouldincreasesignificantly
theapplicabilityoftheproposedscheme. vehicle dynamic model.
7
etm()[]p1−055 −00..011 etm()[]p2−055 −00..011 utkNt()[]1−1310500 utkNt()[]2−1310500
etm()[]p3−055 −00..011 etm()[]p4−055 −00..011 utkNt()[]3−1310500 utkNt()[]4−1310500
etm()[]p5−055 −00..011 etm()[]p6−055 −00..011 utkNt()[]5−1310500 utkNt()[]6−1310500
etm()[]p7−055 −00..011 etm()[]p8−055 −00..011 utkNt()[]7−1310500 utkNt()[]8−1310500
etm()[]p9−0550 −2000..011 40t[sec]60 80 100 12etm()[]0p10−0550 −2000..011 40t[sec]60 80 100 120 utkNt()[]9−13105000 20 40t[sec]60 80 100 1utkNt2()[]010−13105000 20 40t[sec]60 80 100 120
Fig.3. Thepositionerrorsepi(t)(BD). Fig.4. Therequiredcontrolinputsignals(BD).
2x 105 (a) Predecessor−Follower Architecture 2x 107 (b) Bidirectional Architecture the feedback linearization technique deviated up to 15% from
Linear Controller of [9]
Nonlinear Controller of [9] their actual values. Additionally, the corresponding control
1.5 Proposed Controller 1.5
gains were selected through a tedious trial-and-error process
Ets1 1 to yield satisfactory performance for N = 10. Regarding
the proposed control schemes, the parameters were chosen
0.5 0.5
as in Section IV-A, except for the steady state error bound
0 0 and the minimum convergence speed of the performance
x 104 (c) Predecessor−Follower Architecture x 106 (d) Bidirectional Architcture
2 12 functions ρ (t),ρ (t). In particular, ρ was calculated as
pi vi ∞
1.5 10 ρ∞ = 0.5σ√mNin(S),andtheminimumspeedofconvergencewas
8 obtained by the exponential exp(−2t). Finally, the desired
Ess1 6 velocity profile of the leader and the desired inter-vehicular
4 distances were set as in Section IV-A.
0.5
2 The results of the comparative simulation study are given
00 50 N 100 150 00 50 N 100 150 in Figs. 5a-5d. More specifically, Figs. 5a and 5b illustrate
the evolution of E for the predecessor-following and the
ts
Fig.5. TheerrormetricsEts andEss forthePFandBDarchitectures,as bidirectional control architecture respectively. Similarly, the
thenumberofvehiclesN increases. evolution of E is given in Figs. 5c and 5d. Notice that
ss
the proposed control protocols render the metrics E and
ts
E almost invariant to the number of vehicles N. On the
B. Comparative Studies ss
contrary, the performance of the linear and nonlinear control
To investigate further the performance of the proposed
methodologies proposed in [9] deteriorated in both control
methodology, a comparative simulation study was carried out,
architecturesasthenumberofvehiclesincreased,provingthus
onthebasisoftheaforementionednonlinearmodel,amongthe
the superiority of the proposed control protocols.
proposed control schemes and the linear as well as nonlinear
control protocols presented in [9]. For comparison purposes, V. EXPERIMENTALRESULTS
we adopted the following metrics of performance:
To verify the performance of the proposed scheme, an
1 (cid:90) ts(cid:88)N experimental procedure was carried out for the case of the
Ets = N {(ep0,i(t))2+(e˙p0,i(t))2}dt (30) predecessor-followingarchitecture.Theexperimenttookplace
0 i=1 along a 10 m long hallway and lasted approximately 18
1 (cid:90) T (cid:88)N seconds. Five mobile robots were employed. Particularly, a
E = {(e (t))2+(e˙ (t))2}dt (31)
ss N p0,i p0,i Pioneer2AT was assigned as the leading vehicle whereas two
ts i=1 KUKA youBot platforms and two Pioneer2DX mobile robots
for the transient and the steady state respectively, where consistedthefollowingvehicles.Toacquiretheinter-vehicular
e (t), i=1,...,N denote the distance errors with respect distance measurements, infrared proximity sensors operating
p0,i
totheleader,t denotesthetransientperiodandT istheover- from 5 to 65 cm were utilized. The control scheme was
s
all simulation time. In particular, we study through extensive designedatthekinematiclevel,i.e.thecontrolinputswerethe
numericalsimulationshowthemetricsE andE scalewith desiredvelocities(10)sincetheembeddedmotorcontrollerof
ts ss
thenumberofagentsN ∈[10,150]forT =120.Itshouldbe thevehicleswasresponsibleforimplementingtheactualwheel
noticed that the methods proposed in [9] considered a double torque commands that achieved the desired velocities.
integrator model and therefore a feedback linearization tech- The leader adopted a constant velocity model given by
niquewasadoptedinthecontrolschemeinitially.However,to p (t) = 0.3t m and v (t) = 0.3 m/s. The desired inter-
0 0
simulate a realistic scenario, the model parameters adopted in vehicular distances were set at ∆ = ∆(cid:63) = 0.2 m,
i−1,i
8
eetmtm()[]()[]pp13−−00000000........02460246220 5 t[sec]10 17eetmtm()[]()[].5pp24−−00000000........24624600220 5 t[sec]10 17.5 −ptpt,i,...,m()()=14[]−ii100.0000.06....51235500 ∆∆ccooln ∆i−1,Cionn5eCcotlilvisitiyo nC oCnosntsratrianitntt[sec] 10 AAAAggggeeeennnntttt 123417 .5 u(t)[m/s]u(t)[m/s]13000000000.0.0.0.0.0.0.0.23452345........25354555253545550 5 t[sec]10 17u(t)[m/s]u(t)[m/s].425000000000.0.0.0.0.0.0.0.23452345........25354555253545550 5 t[sec] 10 17.5
Fig.6. Thepositionerrorsepi(t),i=1,...,4. Fig.7. Thedistancebetweensuccessivevehicles. Fig.8. Therequiredcontrolinputsignals.
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