Coordinated path-following of multiple underactuated autonomous vehicles with bidirectional communication constraints
A.Pedro Aguiar,Reza Ghabcheloo,Ant´o nio Pascoal,Carlos Silvestre,Jo˜a o Hespanha,and Isaac Kaminer
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Abstract—This paper address the problem of steering a group of underactuated autonomous vehicles along given spatial paths,while holding a desired inter-vehicle formation pattern. For a general class of vehicles moving in either two or three-dimensional space,we show how Lyapunov-bad techniques and graph theory can be brought together to yield a decentral-ized control structure where the dynamics of the cooperating vehicles and the constraints impod by the topology of the inter-vehicle communications network are explicitly taken into account.Path-following for each vehicle amounts to reducing the geometric error to a small neighborhood of the origin. The desired spatial paths do not need to be of a particular ,trimming trajectories)and can be any sufficiently smooth curves.Vehicle coordination is achieved by adjusting the speed of each vehicle along its path according to information on the positions and speeds of a subt of the other vehicles, as determined by the communications topology adopted.We illustrate our design procedure for underwater vehicles moving in three-dimensional space.Simulations results are prented and discusd.
I.I NTRODUCTION
Increasingly challenging mission scenarios and the advent of powerful embedded systems and communication networks have spawned widespread interest in the problem of coordi-nated motion control of multiple autonomous vehicles.The types of applications envisioned are manifold and include aircraft and spacecraft formationflying control[5],[11], [15],coordinated control of land robots[6],[14],and control of multiple surface and underwater vehicles[7],[13],[16]. In spite of significant progress in the area,however,much work remains to be done to develop strategies capable of yielding robust performance of afleet of vehicles in the prence of complex vehicle dynamics,vere communica-tion constraints,and partial vehicle failures.The difficulties are specially challenging in thefield of marine robotics for two main reasons:i)the dynamics of marine vehicles are often complex and cannot be simply ignored or drastically simplified for control design propos,and ii)underwater communications and positioning rely heavily on acoustic sys-Rearch supported in part by the Portugue FCT POSI programme under framework QCA III and by project MAY A-Sub of the AdI.The cond author benefited from a scholarship of FCT.
A.Pedro Aguiar,Reza Ghabcheloo,Ant´o nio Pascoal,and Car-los Silvestre are with the Dept.Electrical Engineering and Com-puters and the Institute for Systems and Robotics,Instituto Su-
perior T´e cnico,Av.Rovisco Pais,1,1049-001Lisboa,Portugal {pedro,reza,antonio,cjs}@isr.ist.utl.pt
Jo˜a o Hespanha is with the Department of Electrical and Computer Engineering,University of California,Santa Barbara,CA93106-9560,USA hespanha@ece.ucsb.edu
Isaac Kaminer is with the Department of Mechanical and Astronauti-cal Engineering,Naval Postgraduate School,Monterey,CA93943,USA kaminer@nps.navy.edu tems,which are plagued with intermittent failures,latency, and multipath effects.
Inspired by the developments in thefield,this paper tackles a problem in coordinated vehicle control that departs slightly from mainstream work reported in the literature. Specifically,we consider the problem of coordinated path-following where multiple vehicles are required to follow pre-specified spatial paths while keeping a desired inter-vehicle formation pattern in time.This problem aris for example in the operation of multiple autonomous underwater vehicles(AUV)for fast acoustic coverage of the abed. In this application,two or more vehicles are required to fly above the abed at the same or different depths,along geometrically similar spatial paths,and map the abed using identical suites of acoustic nsors.By requesting that the vehicles traver identical paths so that the projections of the acoustic beams on the abed exhibit some overlapping, large areas can be c
overed in a short time.The objectives impo constraints on the inter-vehicle formation pattern.A number of other scenarios can of cour be envisioned that require coordinated motion control of marine vehicles.
We solve the coordinated path-following problem for a general class of underactuated vehicles moving in either two or three-dimensional space.The solution adopted is well rooted in Lyapunov-bad theory and address explicitly the vehicle dynamics as well as the constraints impod by the topology of the inter-vehicle communications network. The latter are tackled in the framework of graph theory [12],which ems to be the tool par excellence to study the impact of communication topologies on the performance that can be achieved with coordination[8].The class of vehicles for which the design procedure is applicable is quite general and includes any vehicle modeled as a rigid-body subject to a controlled force and either one controlled torque if it is only moving on a planar surface or two or three independent control torques for a vehicle moving in three dimensional space.Furthermore,contrary to most of the approaches described in the literature,the controller propod does not suffer from geometric singularities due to the parametrization of the vehicle’s rotation matrix.This is possible becau the attitude control problem is formulated directly in the group of rotations SO(3).
With the t-up adopted,path-following(in space)and inter-vehicle coordination(in time)are esntially decoupled. Path-following for each vehicle amounts to reducing a con-veniently defined error variable to zero.The desired spatial paths do not need to be of a particular ,trimming trajectories)and can be any sufficiently smooth curves. Vehicle coordination is achieved by adjusting the speed of
each of the vehicles along its path,according to information on the relative position and speed of the other vehicles,as determined by the communications topology adopted.No other kinematic or dynamic information is exchanged among the vehicles.
This paper builds upon and combine previous results obtained by the authors on path-following control[2],[4] and coordination control[9],[10].
II.P ROBLEM STATEMENT
Consider an underactuated vehicle modeled as a rigid body subject to external forces and torques.Let{I}be an inertial coordinate frame and{B}a body-fixed coordinate frame who origin is located at the center of mass of the vehicle. The configuration(R,p)of the vehicle is an element of the Special Euclidean group SE(3):=SO(3)×R3,where R∈SO(3):={R∈R3×3:RR′=I3,det(R)=+1}is a rotatio
n matrix that describes the orientation of the vehicle by mapping body coordinates into inertial coordinates,and p∈R3is the position of the origin of{B}in{I}.Denoting by v∈R3andω∈R3the linear and angular velocities of the vehicle relative to{I}expresd in{B},respectively, the following kinematic relations apply:
˙p=Rv,(1a)
˙R=RS(ω),(1b) where
S(x):= 0−x3x2
x30−x1
−x2x10
,∀x:=(x1,x2,x3)′∈R3.
We consider here underactuated vehicles with dynamic equa-tions of motion of the following form:
M˙v=−S(ω)M v+f v(v,p,R)+g1u1,(2a)
J˙ω=−S(v)M v−S(ω)Jω+fω(v,ω,p,R)+G2u2,
(2b) where M∈R3×3and J∈R3×3denote constant symmetric positive definite mass and inertia matrices;u1∈R and u2∈R3denote the control inputs,which act upon the system through a constant nonzero vector g1∈R3and a constant nonsingular matrix1G2∈R3×3,respectively;and f v(·),fω(·)reprent all the remaining forces and torques acting on the body.For the special ca of an underwater vehicle,M and J also include the so-called hydrodynamic added-mass M A and added-inertia J A matrices,respectively, i.e.,M=M RB+M A,J=J RB+J A,where M RB and J RB are the rigid-body mass and inertia matrices,respectively. For an underactuated vehicle restricted to move on a planar surface,the same equations of motion(1)–(2)apply without thefirst two right-hand-side terms in(2b).Also,in this ca, (R,p)∈SE(2),v∈R2,ω∈R,g1∈R2,G2∈R, u2∈R,with all the other terms in(2)having appropriate dimensions,and the skew-symmetric matrix S(ω)is given by S(ω)= 0−ωω0 .
For each vehicle,the problem of following a predefined desired path is stated as follows:
1See[4,Remark4]for the special ca of G2∈R3×2.
Path-following problem:Let p d
i
(γi)∈R3be a desired path parameterized by a continuous variableγi∈R and v r
i
(γi)∈R a desired speed assignment for the vehicle i.Suppo also that p d
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i
(γi)is sufficiently smooth and its derivatives(with respect toγi)are bounded.Design a controller such that all the clod-loop signals are bounded, and the position of the vehicle i)converges to and remains inside a tube centered around the desired path that can be made arbitrarily , p i(t)−p d
i
(γi(t)) converges to a neighborhood of the origin that can be made arbitrarily small,and ii)satisfies a desired speed assignment v r
i
along the ,|˙γi(t)−v r
i
(γi(t))|→0as t→∞.
We now consider the problem of coordinated path-following control.In the most general t-up,one is given a t of n≥2autonomous underactuated vehicles and a t of n spatial paths p d
i
(γi);i=1,2,...,n and require that vehicle i follow path p d
i
.As will become clear,the coordination problem will be solved by adjusting the speeds of the vehicles as functions of the“along-path”distances among them.Formally,the along-path distance between vehicle i and j is defined asγij:=γi−γj,and coordination achieved whenγij=0for all i,j∈{1,...,n}[10].
Let J i be the index t of the vehicles that vehicle i com-municates with.Assume that the underlying communication graph is undirected and ,the communication links are bidirectional and there exists a path connecting every two vehicles).In this ca the graph Laplacian L∈R n×n is sym
metric,with a simple eigenvalue at zero and an associated eigenvector1=[1]n×1.The other eigenvalues are positive.See[12]for the definitions and the properties of graphs.The Laplacian can be decompod as L=MM′, where M∈R n×n−1,Rank M′=Rank L=n−1and M′1=0.Define the“graph-induced coordination error”as θ:=Mγ∈R n−1,whereγ:=[γi]n×1.From the properties of M,it can be easily en thatθ=0is equivalent toγi=γj,∀i,j.Conquently,ifθis driven to zero asymptotically, so are the coordination errorsγi−γj and the problem of coordinated path-following is solved.
Coordination problem:Derive a control law for¨γi as a function ofγj and˙γj where j∈J i such thatθapproaches a small neighborhood of zero as t→∞.Each of the n vehicles has access to its own states and exchanges information on its coordination stateγi and speed˙γi with some or all of the other vehicles defined by ts J i.
III.M AIN RESULTS
大气的诗句A.Path-following
形容寒冷的成语In this ction,we briefly discuss the results prented in[2],[4]to solve the path-following problem.Let e i:= R′i p i(t)−p d i(γi(t)) be the path-following error of the vehicle i expresd in its body-fixed frame.Borrowing from the techniques of backstepping,in[2],[4]a feedback law for u1
i
,u2
i
was derived that makes the time-derivative of the Lyapunov function
V i:=
1
2
e′i e i+
1
2
ϕ′i M2iϕi+
1
2
z′2
i
J i z2
i
y [m]
x [m]z [m ]
Fig.1.Coordination of 3AUVs in in-line formation.
take the form
˙V i =−k e i e ′i M −1i e i +e ′i δi −ϕ′i K ϕi ϕi −z ′2i K z 2i z 2i +µi ηi where ϕi and z 2i are linear and angular velocity errors (e [2],[4]for details),k e i ,K ϕi ,K z 2i are positive definite matrices,δi is a small constant vector,and µi captures the terms associated to the speed error ηi :=˙γi −v r i .At this point we remark that if all that is required is to solve a pure path-following problem then one can augment V i with the
quadratic term 12η2
i and utilize the freedom of assigning a
feedback law to ¨γi in order to make ˙V
i negative definite (e details in [2],[4]).This strategy must be modified to address coordination as shown below.B.Coordinated path-following
This ction prents a solution to the coordinated path-following problem.Let η:=˙γ−v L 1be the speed vector error,where v L is a desired speed profile assigned to the formation.Consider the composite (coordination +path-following)Lyapunov function
V c :=12θ′θ+12z ′
z +n i =1
V i
where z :=η+A −11µ+A −1
1Mθ.Computing the time-derivative of V c and assigning the following feedback law for ¨
γ¨γ=−A −1
圣诞壁纸1˙µ
−A 1η−A −11Lη−A 2z,(3)where A 1,A 2are diagonal positive definite matrices,we obtain
˙V c =−η′A 1η−z ′A 2z −n i =1
k e i e ′i M −1i e i −e ′i δi
+ϕ′i K ϕi ϕi
+
z ′
2i K z 2i z 2i .
教务处工作职责It is now straightforward to prove the following result:Theorem 1:The feedback laws for u 1i ,u 2i for each
vehicle i obtained in [2],[4]together with (3)solve the coordination and the path-following problems.
IV.A N ILLUSTRATIVE EXAMPLE
This ction illustrates the application of the previous results to underwater vehicles moving in three-dimensional space.
A.Path-following and coordination of underwater vehicles in 3-D space
Consider an ellipsoidal shaped underactuated autonomous underwater vehicle (AUV)not necessarily neutrally buoyant.Let {B}be a body-fixed coordinate frame who origin is located at the center of mass of the vehicle and suppo that we have available a pure body-fixed control force τu in the x B direction,and two independent control torques τq and τr about the y B and z B axes of the vehicle,respectively.The kinematics and dynamics equations of motion of the vehicle can be written as (1)–(2),where
M =diag {m 11,m 22,m 33},u 1=τu
J =diag {J 11,J 22,J 33},
u 2=(τq ,τr )′
D v (v )=diag {X v 1+X |v 1|v 1|v 1|,Y v 2+Y |v 2|v 2|v 2|,
Z v 3+Z |v 3|v 3|v 3|}
D ω(ω)=diag {K ω1+K |ω1|ω1|ω1|,M ω2+M |ω2|ω2|ω2|,
N ω3+N |ω3|ω3|ω3|}
g 1=
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,
1号文件
G 2=
001001
,
¯g 1(R )=R
′
0W −B
,
¯g 2(R )=S (r B )R
′
00
B
time [s]
Fig.2.Time evolution of the coordination errors γ12:=γ1−γ2and γ13:=
Fig.3.Time evolution of the path-following errors p i −p d i ,i =1,2,3.
f v =−D v (v )v −¯
g 1(R ),f ω=−D ω(ω)ω−¯g 2(R ).
The gravitational and buoyant forces are given by W =mg and B =ρg ∇,respectively,where m is the mass,ρis the mass density of the water and ∇is the volume of displaced water.The numerical values ud for the physical parameters match tho of the Sirene AUV ,described in [1],[3].B.Simulation results
This ction contains the results of simulations that illus-trate the performance obtained with the coor
dinated path-following control laws developed in the paper.Figures 1–3illustrate the situation where three underactuated AUVs are required to follow paths of the form
p d i (γi )= a 1cos(2πT γi +φd ),a 1sin(2π
T
γi +φd ),a 2γi +z 0i ,
with a 1=20m ,a 2=0.05m ,T =400,φd =−3π
4,and z 01=−10m,z 02=−5m,z 03=0m .The initial conditions of the AUVs are p 1=(x 1,y 1,z 1)=(10m,−10m,−5m ),p 2=(x 2,y 2,z 2)=(5m,−15m,0m ),p 3=(x 3,y 3,z 3)=(0m,−20m,5m ),R 1=R 2=R 3=I ,and v 1=v 2=v 3=ω1=ω2=ω3=0.The vehicles are required to keep a formation pattern whereby they are aligned along a vertical line.In the simulation,vehicle 1is allowed to communicate with vehicles 2and 3,but the last two do not communicate between themlves directly.The reference speed v L was t to v L =0.5s −1.Notice how the vehicles adjust their speeds to meet the formation requirements.Moreover,the coordination errors γ12:=γ1−γ2and γ13:=γ1−γ3and
the path-following errors converge to a small neighborhood of the origin.
V.C ONCLUSIONS
The paper addresd the problem of steering a group of underactuated autonomous vehicles along given spatial paths,while holding a desired inter-vehicle formation pattern (coordinated path-following).A solution was derived that builds on recent results on path-following control [2],[4]and state-agreement (coordination)control [9],[10]obtained by the authors.The solution propod builds on Lyapunov bad techniques and address explicitly the constraints impod by the topology of the inter-vehicle communications network.Furthermore,it leads to a decentralized control law whereby the exchange of data among the vehicles is kept at a minimum.Simulations illustrated the efficacy of the solution propod.Further work is required to extend the methodology propod to address the problems of robustness against temporary communication failures.
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