DTIC ADA439756: Decentralized Control of Autonomous Vehicles

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T R R
ECHNICAL ESEARCH EPORT
Decentralized Control of Autonomous Vehicles
by John S. Baras, Xiaobo Tan, Pedram Hovareshti
CSHCN TR 2003-8
(ISR TR 2003-14)
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Decentralized Control of Autonomous Vehicles
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Decentralized Control of Autonomous Vehicles∗
John S. Baras, Xiaobo Tan, and Pedram Hovareshti
Institute for Systems Research and
Department of Electrical and Computer Engineering
University of Maryland, College Park, MD 20742 USA
{baras, xbtan, hovaresp}@glue.umd.edu
Abstract
Decentralized control methods are appealing in coordination of multiple vehicles due to their low
demand for long-range communication and their robustness to single-point failures. An important
approachin decentralizedmulti-vehicle controlinvolvesartificialpotentials or digital pheromones. In
this paper we explore a decentralized approach to path generation for a group of combat vehicles in
a battlefield scenario. The mission is to maneuver the vehicles to cover a target area. The vehicles
are required to maintain good overall area coverage, and avoid obstacles and threats during the
maneuvering. The gradient descent method is used, where each vehicle makes its moving decision
by minimizing a potential function that encodes information about its neighbours, obstacles, threats
and the target. We conduct analysis of vehicle behaviors by studying the vector field induced by the
potential function. Simulation has shown that this approachleads to interesting emergentbehaviors,
and the behaviors can be varied by adjusting the weighting coefficients of different potential function
terms.
1 Introduction
Autonomous unmanned vehicles (AUVs) are receiving tremendous interest dueto their potentially revo-
lutionizingapplicationsindefense,transportation,weatherforecast,andplanetaryexploration[1]. These
vehicles are often deployed in groups to perform complicated missions. Communication is often limited
in theseapplications dueto the large numberof vehicles involved, limited battery power, and constraints
imposed by environmental conditions or mission requirements. Hence a decentralized approach to co-
∗ThisresearchwassupportedbytheArmyResearchOfficeundertheODDR&EMURI01ProgramGrantNo. DAAD19-
01-1-0465 to theCenter for Networked Communicating Control Systems (through Boston University).
1
ordination and control of multi-vehicles is especially appealing. A decentralized method has another
advantage over a centralized one: it is more robust to the problem of single-point failure.
Inspiredbytheemergentbehaviorsdemonstratedbyswarmsofbacteria, insects,andanimals,control
methods which yield desired collective behaviors based on simple local interactions are of great interest
[2, 3, 4]. Artificial potentials or digital pheromones are typically involved in such methods for multi-
vehicle control, see e.g., [3, 5, 6, 4] and the references therein. The potential function method has been
used in various robotic applications [7]. The idea is to derive a force or other input (e.g.,velocity) from
some potential function which encodes relevant information about the environment and the mission.
In this paper we explore a decentralized approach to path generation for a group of combat vehicles
in a battlefield scenario using the potential function method. The mission is to maneuver the vehicles
to cover a target area. The vehicles are also expected to maintain good overall area coverage during the
maneuvering, and avoid obstacles and threats. At every time instant each vehicle evaluates its potential
function profile and decides its velocity based on the gradient descent method. The potential function
consists of several terms reflecting the objectives and the constraints. It is constructed in such a way
that only information about neighbouring vehicles, local information about dynamic threats, and some
static information (about stationary threats, targets) are involved.
We analyze vehicle formations at equilibria by taking the sum of potential functions of all vehicles
as a Lyapunov function candidate. We also study the behavior of a vehicle experiencing both attraction
from the target and repulsion from the obstacles by analyzing its vector field. Simulation is performed
in Matlab, and it shows that the decentralized approach leads to interesting emergent behaviors, and
the behaviors can be varied by adjusting the weighting coefficients of diffrent potential function terms.
The remainder of the paper is organized as follows. In Section 2 we describe the problem setup and
construct the potential functions. We perform analysis of vehicle behaviors in Section 3. Simulation
results are reported in Section 4. Section 5 concludes the paper.
2 Potential Functions
WestudythekinematicplanningproblemforN vehiclesmovingona(twodimensional)plane. Extension
to three dimensional space is straightforward, although the analysis will be more complicated. Each
vehicle is treated as a point. The coordinates of the i-th vehicle V is denoted as p = (x ,y ), 1≤ i ≤ N.
i i i i
The task for the vehicles is to move toward and then occupy a connected target area A ⊂ R2. They
should avoid to crash into obstacles that are distributed in the battlefield. There are also threats, both
2
stationary ones and moving ones, that endanger the vehicles if they are close. We assume that each
vehicle has the knowledge of locations of stationary threats, and it can detect a moving threat if the
threat is within the distance R .
m
The vehicles can talk to each other and exchange information about their positions and velocities if
they are within the neighbouring distance R . Let V(t) be the set of vehicles alive at t. For V ∈ V(t),
c i
we define its neighbouring set
(cid:1)
N(V )= {V ∈ V(t) :(cid:5) p −p (cid:5)≤ R }.
i j i j c
If two or more vehicles get too close, there is a chance of collision. That also makes it easier for the
enemy fire to target the vehicles. Another disadvantage of being too close is that the overall coverage
area is small. On the other hand, if vehicles are too far apart, they lose contact. Therefore, there is an
optimal distance r0 < Rc between two vehicles.
From the above discussions, there are multiple objectives/constraints when a vehicle makes the
moving decision. To accomodate that we construct a potential function J¯i for each vehicle V at time t,
t i
where J¯i consists of several terms, each term reflecting a goal or a constraint. To be specific,
t
J¯ti = λgJg(pi(t))+λnJn(pi(t),{pj(t)}j(cid:2)=i)+λoJo(pi(t))+λsJs(pi(t))+λmJm(pi(t),t), (1)
where Jg,Jn,Jo,Js,Jm are components of the potential function relating to the target, neighbouring
vehicles, obstacles, stationary threats, and moving threats, respectively, and λ ,λ ,λ ,λ , and λ ≥ 0
g n o s m
are weighting coefficients. The velocity of the vehicle V is specified by
i
∂J¯i
p˙ (t) = − t. (2)
i ∂p
i
We now describe in detail the components of the potential function.
• The target potential Jg. Denote ρ(p,A) = infa∈A (cid:5) p−a (cid:5), the distance from the point p to the
target area A. We then let
Jg(p ) = fg(ρ(p ,A)),
i i
where fg(·) is a strictly increasing function, and fg(0) = 0. This guarantees that in the absence of
other objects, the vehicle will move toward the target. For analysis and simulation in this paper,
we choose fg(r)= r2.
• The neighbouring potential Jn. Since distance (as opposed to direction) is our concern here, we
let
(cid:1)
Jn(pi(t),{pj(t)}j(cid:2)=i) = fn((cid:5) pi(t)−pj(t)(cid:5)), (3)
j(cid:2)=i:Vj∈N(Vi)
3
25
20
ntial15
ote10
P
5
0
1 2 3 4 5 6 7 8 9
Neighbouring distance
ve 5
vati 0
deri −5
ntial −10
ote−15
P
−20
1 2 3 4 5 6 7 8 9
Neighbouring distance
Figure 1: An example of the neighbouring potential function.
where fn : R+ → R is a differentiable function that has the following properties: a) fn(r) ap-
proaches infinity as r → 0, and is strictly deceasing in [0,r0]; b) it is strictly increasing in [r0,Rc]
and dfn(R ) = 0. These properties enable two vehicles to keep the optimal distance in the absence
dr c
of other objects, and make the dynamicstransition seamless whenthe neighbouringset of avehicle
is changing. An example of such fn and its derivative is shown in Figure 1, where r0 = 2, Rc = 8.
• The obstacle potential Jo. An obstacle is a connected, closed set (could be a single point) that a
vehicle cannot enter. We assume that there are a finite number of obstacles {O }No . Then
j j=1
(cid:1)No
Jo(p ) = fo(ρ(p ,O )), (4)
i i j
j=1
where ρ(p ,O ) is the distance from p to the set O , and fo(·) : R+ → R is a strictly decreasing
i j i j
function and fo(r) → ∞ as r → 0. One example of fo is fo(r) = 1 .
r2
• The potential Js due to stationary threats. Stationary threats can be modeled similarly as ob-
stacles, so that vehicles will avoid to get close to them. Anisotropic threats (dangers that are
direction-dependent) can be taken care of using appropriate potential functions.
• Thepotential Jm duetomovingthreats. Amoving threatisamovingpointmass. Thei-th vehicle
is able to see the moving threat M if (cid:5) p −q (cid:5)≤ R , and is killed by M if (cid:5)p −q (cid:5)≤ R < R ,
j i j d j i j e d
where q denotes the position of M . Let M (t) be the set of moving threats in the i-th vehicles’s
j j i
detection range, then we let
(cid:1)
Jm(p ,t) = fm((cid:5) p −q (cid:5)), (5)
i i j
j:Mj∈Mi(t)
where the function fm : (R ,∞) → R is differentiable, strictly decreasing on (R ,R ), constant on
e e d
(R ,∞), and fm(r) → ∞ when r → R . One can see that with this potential function, a vehicle
d e
4
tries to keep at least a distance R from moving threats, and its vector field remains continuous
e
when moving threats enter or leave its detection range. A simple example for such fm(·) is

 (r−1Re)2 if Re < r ≤ Re+2Rd
fm(r) = 16(r−Rd)2 − 8Re if Re+Rd ≤ r ≤ R
 (Rd−Re)3(Rd+Re) (Rd−Re)3 2 d
− 8Re if r > R
(Rd−Re)3 d
3 Qualitative Analysis of Vehicle Behaviors
In this section, we analyze vehicle behaviors under the gradient descent method. In particular, we
analyze how vehicles settle down after they enter the target area, and study the behavior of a vehicle
when it experiences both attraction from the target and the repulsion from the obstacles.
3.1 Stability of equilibrium configurations
We firstconsider multi-vehicles insidethe target area. We are interested in knowingwhetherthevehicles
will settle down for an equilibrium configuration under interactions, and if so, whether the equilibrium
configuration is stable. Herewe assumethattheonly componentof thepotential function ofeach vehicle
is Jn.
Proposition 3.1 Let N be the number of vehicles.
1. For any N, the configuration of vehicles converges to an equilibrium under interactions.
2. For N =2, the vehicles maintain a distance of r0 in the equilibrium configuration and the equilib-
rium is globally stable.
3. For N = 3, assuming that dfn is strictly increasing in (0,r0], there are two possible equilibrium
dr
configurations, equilateral triangular withspacing r0 (Figure 2(a)), and collinear with equalspacing
r(cid:4) (Figure 2 (b)), where r20 < r(cid:4) <r0 and
dfn dfn
(r(cid:4))= − (2r(cid:4)). (6)
dr dr
If dfn is strictly increasing in [r0,2r0], r(cid:4) is unique. The collinear configuration is unstable, while
dr
the equilateral configuration is stable.
5
r r r r
0 0
r
0
(a) (b)
Figure 2: Equilibrium configurations for N =3.
Proof. 1. Define a candidate Lyapunov function
(cid:1)N
1
J({pi(t)}) = Jn(pi(t),{pj(t)}j(cid:2)=i).
2
i=1
It’s easy to verify that
dJ (cid:1)N ∂Jn
dt = − (cid:5) ∂p (pi,{pj}j(cid:2)=i) (cid:5)2 . (7)
i=1 i
Hence J is nonincreasing with t. Since J is lower bounded, dJ → 0. This implies that p˙ (t) → 0, ∀i, and
dt i
hence the vehicles converge to an equilibrium configuration.
2. When N = 2, J(p1(t),p2(t)) = fn((cid:5) p1(t)−p2(t)(cid:5)). Writing r12(t) =(cid:5)p1(t)−p2(t) (cid:5), we have
dJ dfn
dt = −2( dr (r12(t))2.
Hencer12(t) → r0 ast → ∞,andit’sclearthatthisisastableconfiguration(aslongas(cid:5)p1(0)−p2(0) (cid:5)<
R ).
c
3. When N = 3,
J(p1(t),p2(t),p3(t)) = fn(r12(t))+fn(r23(t))+fn(r31(t)),
where r (t) =(cid:5)p (t)−p (t) (cid:5). Then dJ =
ij i j dt
dfn dfn dfn dfn dfn dfn
− (cid:5) dr (r12)ˆr12+ dr (r31)ˆr13) (cid:5)2 − (cid:5) dr (r12)ˆr21+ dr (r23)ˆr23) (cid:5)2 − (cid:5) dr (r31)ˆr31+ dr (r23)ˆr32)(cid:5)2,
(8)
where ˆr denotes the unit vector pointing from p to p . Eq. (8) implies J is strictly decreasing unless
ij j i
either
• dfn(r12) = dfn(r23) = dfn(r31) = 0, which corresponds to the equilateral triangular configuration
dr dr dr
shown in Figure 2 (a), or
• rˆij’s are parallel or antiparallel and, assuming p2 is between p1 and p3 (we do not lose generality
since three vehicles have symmetric roles), r12 = r23 = r(cid:4) for r(cid:4) ∈ (r20,r0) satisfying (6), which
corresponds to the collinear configuration in Figure 2(b).
6
y
Target
x
Obstacle 1 Obstacle 2
(−a,−b) (a,−b)
Vehicle
Figure 3: The setup of two obstacles and one target
The equilateral triangular configuration is stable. Indeed, for any perturbation with the corre-
sponding J ∈ (3fn(r0),2fn(r(cid:4))+fn(2r(cid:4))), the system will come back to this configuration. The
collinear configuration isnotunstable, sincesmallperpendicularperturbationof themiddlevehicle
leadstostrictdecreaseofJ andthesystemwillconvergetotheequilateraltriangularconfiguration.
(cid:1)
Remark 3.1 We note that similar results appeared in [6] where the second order dynamics of point
masses was considered.
3.2 Vector field analysis
First we consider the scenario as shown in Figure 3. The target is located at the origin (0,0). There
are two (point) obstacles located symmetrically about the y axis with coordinates (−a,−b) and (a,−b),
respectively, where a,b > 0. The potential function in terms of (x,y) is taken to be
1 1
λ (x2+y2)+ + ,
g (x+a)2+(y+b)2 (x−a)2+(y+b)2
and the associated vector field is

 x˙(t) = 2(x+a) + 2(x−a) −2λ x
[(x+a)2+(y+b)2]2 [(x−a)2+(y+b)2]2 g . (9)
 y˙(t) = 2(y+b) + 2(y+b) −2λ y
[(x+a)2+(y+b)2]2 [(x−a)2+(y+b)2]2 g
We consider a vehicle on the x axis, and study whether it will move toward the target under the
vectorfield(9)wheny < 0(thecasey > 0issimplerandcanbestudiedsimilarly). Duetothesymmetry,
x˙ = 0, so the real question is whether y˙ > 0. When x = 0,
4(y+b)
y˙ = −2λ y. (10)
[(a2+(y+b)2]2 g
7
y
Target
x
Obstacle 1 Obstacle 2
(−a,−b) (a,−b)
y*
1
y*
y*
2
Figure 4: Vector field on the x-axis for the case of two obstacles and one target.
Let y˜= y+b. Obviously, if y˜≥ 0, y˙ > 0. In the following we study the case y˜< 0, i.e., y < −b.
Proposition 3.2 There is a unique solution y˜∗ ∈ (−√a ,0) to
3
4y˜3−3by˜2+a2b = 0. (11)
Let
2 4y˜∗2
λ∗ = (1− ).
g (a2+y˜∗2)2 a2+y˜∗2
Let y∗ = y˜∗−b. Then
• If λ > λ∗, y˙ > 0, ∀y < −b;
g
• If λ = λ∗, y˙ > 0 for y ∈ (−∞,−b) except at y∗ where y˙ = 0;
g
• If λ < λ∗, there exist y∗,y∗ such that y∗ < y∗ < y∗, and
g 1 2 2 1

 y˙ > 0, if y ∈ (−∞,y2∗)


y˙ < 0, if y ∈ (y∗,y∗)
2 1 ,
 y˙ > 0, if y ∈ (y1∗,−b)

 y˙ = 0, if y = y∗ or y∗
1 2
as illustrated in Figure 4. Furthermore, y∗ (y∗, resp.) increases (decreases, resp.) with λ and
1 2
y∗ → −b, y∗ → −∞ as λ → ∞.
1 2
Proof. Let
4y˜
h(y˜)= .
(a2+y˜2)2
8