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Evolutionary Advantage of
Reciprocity in Collision Avoidance
Daniel Hennes, Daniel Claes, and Karl Tuyls
Department of Knowledge Engineering, Maastricht University
P.O. Box 616, 6200MD Maastricht, The Netherlands
{daniel.hennes,daniel.claes,k.tuyls}@maastrichtuniversity.nl
Abstract. Collision avoidance is a complex task, especially in the pres-
ence of dynamic obstacles. The task increases in complexity when the
dynamic obstacles are mobile robots that also take actions to avoid colli-
sions. On the other hand, assuming mutual avoidance (reciprocity), can
improve avoidance behavior since each robot only takes half of the re-
sponsibility of avoiding pairwise collisions. This paper combines research
in evolutionary game theory and multi-robot collision avoidance to an-
alyze the stability of various velocity obstacle based collision avoidance
methods in competition. Results show that reciprocity is advantageous
under evolutionary dynamics.
Keywords: Multi-robot collision avoidance, evolutionary game theory,
heuristic payoff tables, replicator dynamics
1 Introduction
Collision avoidance is relevant for a variety of domains and applications, e.g.,
crowd dynamics as well as automobile, aircraft, vessel, or multi-robot collision
avoidance systems. Each of the above is of high importance in everyday-life, and
of great theoretical and practical interest. Crowd dynamics [6] is the study of
pedestrian motion and how individuals affect each others’ movements locally. A
better understanding of crowd dynamics makes it possible to accurately simulate
emergency and evacuation scenarios, resulting in improved guidelines to prevent
blockages and jamming in case of a panic stampede.
Driver assistance or fully autonomous cars are further examples of appli-
cations requiring collision avoidance systems. Aircraft and vessel traffic follows
predefined sets of rules to avoid collisions or resolve such situations. Automo-
bile, aircraft and vessel traffic are application areas of collision avoidance where
a game-theoretic analysis has been applied in the past [8, 12, 17].
Finally, with the expected increase in the number of robots deployed in fac-
tories as well as in home environments, there is a need for methods to avoid
collision situations in a variety of multi-robot systems [11, 15].
In this paper we analyze the evolutionary stability of various local collision
avoidance methods in simulation. Local collision avoidance is the task of steering
free of collisions with static and dynamic obstacles, while following a global plan
2 Daniel Hennes, Daniel Claes and Karl Tuyls
Table 1. The game of chicken payoff table.
Swerve Straight
Swerve 0, 0 −1, 1
Straight 1,−1−10,−10
to navigate towards a goal location. Static obstacles can be avoided using tra-
ditional planning algorithms whereas dynamic obstacles pose a tough challenge.
An intuitive approach is to observe consecutive obstacle positions in order to
extrapolate the future trajectory. The velocity obstacle [3] is a geometric rep-
resentation of all velocities that will eventually result in a collision given that
the dynamic obstacle maintains the observed velocity. Velocity obstacles find
application in robotics [15, 13, 7, 1, 2] and have also been applied to the study of
crowd dynamics [5].
The remainder of the paper is organized as follows. In Section 2 we introduce
a stylized game example of collision avoidance and evolutionary game theory.
Section 3 covers multi-robot collision avoidance based on the velocity obstacle
paradigm. In particular, we discuss velocity obstacles, reciprocal velocity obsta-
cles and hybrid velocity obstacles. In Section 4 we explain the application of
evolutionary game theory to collision avoidance and discuss the resulting dy-
namics. Finally, Section 5 concludes the paper.
2 Background
In this section we introduce the concepts of evolutionary game theory using a
stylized example, which relates to the situation of collision avoidance: the game
of chicken [10]:
The game of chicken: Two drivers are headed at high speeds for a narrow
passage from opposite directions. If both drivers continue to drive straight
the result is a catastrophic head-on collision. Whoever swerves is considered
a ”chicken” and yields the way to the other driver. The best outcome for a
driver is thus continuing straight while the other swerves and is the “chicken”
thereby successfully avoiding a collision.
We assume identical driver reaction times and turning radii of the cars. There-
fore, this strategic situation occurs in the last instant before a crash is unavoid-
able. Each driver faces the choice of continuing straight or swerving to the side.
The decisions are taking simultaneously and can not be revoked. The payoffs
for the game of chicken are shown in Table 1. There are two pure equilibria in
the game of chicken, “straight–swerve” with payoffs 1,−1 and “swerve–straight”
with payoffs: −1,1. Neither player has a dominant strategy as each player’s best
strategy depends on the strategy played by the adversarial. In addition, there is
Evolutionary Advantage of Reciprocity in Collision Avoidance 3
0 11
x1
Fig. 1. Symmetric replicator dynamics of the game of chicken. State x1= 0 corre-
sponds to a population of “swerve-only”, while x1= 1 corresponds to a population of
“straight-only”. State x= ( 9
10 ,1
10 ) is the unique asymptotically stable fixed point of
the dynamics.
one symmetric mixed Nash equilibrium where both players play “straight” with
probability 9
10 and “swerve” with probability 1
10 .
2.1 Evolutionary game theory
Traditional game theory assumes perfectly rational and self-interested players.
Consequently, classical game theory aims at finding an optimal strategy that
maximizes the expected utility for a player. In contrast, evolutionary game the-
ory is a descriptive approach. A game is played repeatedly by boundedly rational
players with little or no knowledge of the game. Two randomly matched indi-
viduals play preassigned pure strategies according to their phenotype. In evolu-
tionary game theory, the payoff determines the fitness (value of success) of the
represented strategy, or phenotype. The evolutionary process is modeled using
biological-inspired operators, i.e., natural selection, replication and mutation.
Two core concepts of evolutionary game theory are the replicator dynamics and
evolutionarily stable strategies.
The continuous-time replicator dynamics [14] formally define the population
change over time. An infinite population state is represented by a probability
distribution xover all phenotypes (pure strategies). The payoff function fican
be interpreted as the Darwinian fitness of phenotype i.
˙xi=xi
fi(x)−X
j
xjfj(x)
(1)
Evolutionarily stable states [9] are such population distributions that are fixed
points of the replicator dynamics, i.e., ˙x= 0, and where small perturbations
|ˆx−x|< would be driven back to xby selection pressure, i.e., by following the
replicator dynamics.
Figure 1 shows the continuous-time replicator dynamics in the game of chicken
with payoff values as shown in Table 1. All interior points converge to the mixed
Nash equilibrium, which is an evolutionarily stable strategy.
The continuous-time replicator dynamics model a population of individu-
als that are matched up at random and play a one-shot game. The dynamics
shown in Figure 1 are thus the result of the same collision avoidance interaction
occurring over and over with different random drivers.
4 Daniel Hennes, Daniel Claes and Karl Tuyls
3 Velocity Obstacles
Clearly, the game of chicken presents a very abstract (and radical) situation.
Collision avoidance in real domains requires more elaborate decision making than
simply choosing between driving straight or swerving to the side. One approach
to collision avoidance in continuous spaces is the velocity obstacle paradigm. The
velocity obstacle (VO) was first introduced by [3] for local collision avoidance
and navigation in dynamic environments with multiple moving objects. The
subsequent definition of the VO assumes planar motions, though the concept
extends to three dimensional motions in a straightforward manner.
Let us assume a workspace configuration with two robots on a collision course
as shown in Figure 2(a). If the position and speed of the moving object (robot
RB) is known to RA, we can mark a region in the robot’s velocity space which
leads to collision under current velocities and is thus unsafe. This region resem-
bles a cone with the apex at RB’s velocity vB, and two rays that are tangential
to the convex hull of the Minkowski sum of the footprints of the two robots. The
Minkowski sum for two sets of points Aand Bis defined as:
A⊕B={a+b|a∈A, b ∈B}(2)
We define the ⊕operator to denote the convex hull of the Minkowski sum such
that A⊕Bresults in the points on the convex hull of the Minkowski sum of A
and B. In the example, the two robots have circular footprints with radii rAand
rBrespectively.
The direction of the left and right ray is then defined as:
θlef t = max
pi∈FA⊕FB
atan2((prel +pi)⊥·prel ,(prel +pi)·prel )
θright = min
pi∈FA⊕FB
atan2((prel +pi)⊥·prel ,(prel +pi)·prel )
where prel =pB−pAis the relative position of the two robots and FA⊕ FBis
the convex hull of the Minkowski sum of the footprints of the two robots. The
atan2 expression computes the signed angle between two vectors. The resulting
angles θlef t and θright are left and right of prel.
In the example in Figure 2, robot RA’s velocity vector vApoints into the VO,
thus we know that RAand RBare on collision course. Each robot computes a
VO for each of the other robots. If all robots at any given time step select
velocities outside of the VOs, the trajectories are guaranteed to be collision free.
However, oscillations can still occur when the robots are on collision course. Since
all robots select a new velocity outside of all velocity obstacles independently,
at the next time step, the old velocities pointing towards the goal will become
available again. Hence, all robots select their old velocities, which will be on
collision course again after the next time step.
To overcome these oscillations, the reciprocal velocity obstacle (RVO) was
introduced by [15]. The surrounding moving obstacles are in fact also pro-active
robots and thus aim to avoid collisions too. Assuming that each robot takes care
Evolutionary Advantage of Reciprocity in Collision Avoidance 5
x
y
rA
rB
pA
pB
vA
vB
(a) Workspace configuration
vx
vy
rA+rB
pB−pA
vA
vB
(b) VO
(vA+vB)
2
vx
vy
vA
vB
(c) RVO
vx
vy
vA
vB
(d) HRVO
Fig. 2. Creating the different velocity obstacles out of a workspace configuration. (a)
A workspace configuration with two robots RAand RB. (b) Translating the situation
into the velocity space and the resulting velocity obstacle (VO) for RA. (c) Translating
the VO by vA+vB
2results in the reciprocal velocity obstacle (RVO), i.e., each robot
has to take care of half of the collision avoidance. (d) Translating the apex of the RVO
to the intersection of the closest leg of the RVO to the own velocity, and the leg of
the VO that corresponds to the leg that is furthest away from the own velocity. This
encourages passing the robot on a preferred side, i.e., in this example passing on the
left. The resulting cone is the hybrid velocity obstacle (HRVO).
of half of the collision avoidance, the apex of the VO can be translated to vA+vB
2.
Furthermore, this leads to the property that if every robot chooses a velocity
outside of the RVO closest to the current velocity, the robots will pass on the
same side. However, each robot optimizes its commanded velocity with respect
6 Daniel Hennes, Daniel Claes and Karl Tuyls
rA+rB
τ
vx
vy
(a) Truncated VO
vx
vy
vA
vB
(b) Truncated HRVO
Fig. 3. (a) Truncation of a VO of a static obstacle at τ= 2 and approximating the
truncation by a line. (b) Translating the truncated cone according to the HRVO method
to get a truncated HRVO.
to a preferred velocity in order to make progress towards its goal location. This
can lead to reciprocal dances, i.e., where both robots first try to avoid to the
same side and then to the other side. In a situation with perfect symmetry and
sensing, this behavior continues indefinitely.
To counter these situations, the hybrid velocity obstacle (HRVO) was intro-
duced by [13]. Figure 2(d) shows the construction of the HRVO. To encourage
the selection of a velocity towards the preferred side, e.g. left in this example,
the other leg of the RVO is substituted with the corresponding leg of the VO.
The new apex is the intersection of the line of the one leg from RVO and the
line of the other leg from the VO. This reduces the chance of selecting a velocity
on the “wrong” side of the velocity obstacle and thus the chance of a reciprocal
dance, while not overconstraining the velocity space. The robot might still try
to pass on the “wrong” side, e.g., another robot induces a HRVO that blocks
the whole side, but then soon all other robots will adapt to the new side too.
3.1 Truncation
When the workspace is cluttered with many robots that do not move or only
move slowly, the apices of the HRVOs are close to the origin in velocity space;
thus rendering robots immobile. This problem can be solved using truncation.
The idea of a truncated hybrid velocity obstacle can be best explained by imag-
ining a static obstacle. A velocity in the direction of the obstacle will eventually
lead into collision, but not directly. Hence, we can define an area in which the
selected velocities are safe for at least τtime steps. The truncation is then in the
shape of the Minkowski sum of the two footprints, shrunk by the factor τ. If the
footprints are discs, the shrunken disc that still fits in the truncated cone has a
radius of rA+rB
τ, see Figure 3(a). The truncation can be closely approximated
by a line perpendicular to the relative position and tangential to the shrunken
disk. Applying the same method to create a HRVO from a VO, we can create a
Evolutionary Advantage of Reciprocity in Collision Avoidance 7
vx
vy
vA
Fig. 4. ClearPath enumerates intersection points for all pairs of VOs (solid dots). In
addition the preferred velocity vAis projected on the closest leg of each VO (open
dots). The point closest to the preferred velocity (dashed line) and outside of all VOs
is selected as new velocity (solid line).
truncated HRVO out of the truncated VO by translating the apex accordingly,
see Figure 3(b). The same applies to RVOs.
3.2 ClearPath
To efficiently compute collision free velocities, we employ the ClearPath algo-
rithm introduced by [4]. The algorithm is applicable to many variations of ve-
locity obstacles (VO, RVO or HRVO) represented by line segments or rays.
ClearPath follows the general idea that the collision free velocity that is closest
to preferred velocity is: (a) on the intersection of two line segments of any two
velocity obstacles, or (b) the projection of the preferred velocity onto the clos-
est leg of each velocity obstacle. All points that are within another obstacle are
discarded and from the remaining set the one closest to the preferred velocity is
selected. Figure 4 shows the graphical interpretation of the algorithm.
4 Evolutionary Analysis
We have introduced three variations of the velocity obstacle approach, namely
VO, RVO, and HRVO. Evaluating the performance of these methods, especially
in a heterogenous setting, is the aim of this paper. We perform an evolutionary
analysis based on heuristic payoff tables [16] to approximate an infinite popula-
tion. The heuristic payoff table Hcaptures the payoff information for all possible
discrete distributions Nifor a finite population with nindividuals. The payoff
for an arbitrary continuous population state xis computed as the weighted av-
erage over all rows of the heuristic payoff table, where payoffs are weighted by
the probability that the discrete distribution of a particular row Niis the result
of drawing nindividuals according to x. For further details, see Section 4.1.
8 Daniel Hennes, Daniel Claes and Karl Tuyls
4.1 Method
The evolutionary model assumes an infinite population. We cannot compute the
payoff for such a population directly, but we can approximate it from evaluations
of a finite population.
All possible distributions over kinformation levels can be enumerated for
a finite population with nindividuals. Let Nbe a matrix, where each row Ni
contains one discrete distribution. The matrix will yield n+k−1
nrows. Each
distribution over information levels can be simulated with the market model,
returning a vector of average expected relative market revenues u(Ni). Let U
be a matrix which captures the revenues corresponding to the rows in N, i.e.,
Ui=u(Ni). A heuristic payoff table H= (N, U ) is proposed in [16] to capture
the payoff information for all possible discrete distributions in a finite population.
In order to approximate the payoff for an arbitrary mix of strategies xin an
infinite population distributed over the phenotypes according to x,nindividuals
are drawn randomly from the infinite distribution. The probability for selecting
a specific row Nican be computed from xand Ni:
P(Ni|x) = n
Ni,1, Ni,2, . . . , Ni,kk
Y
j=1
xNi,j
j
The expected payoff fi(x) is computed as the weighted combination of the payoffs
given in all rows, compensating for payoff that cannot be measured. If a discrete
distribution features zero traders of a certain information type, its payoffs cannot
be measured and Uj,i = 0.
fi(x) = PjP(Nj|x)Uj,i
1−(1 −xi)k
This expected payoff can be used in (1) to compute the evolutionary change
according to the replicator dynamics.
4.2 Experimental Setup
To compute the payoffs corresponding to each finite population Niwe consider
the following scenario. All robots have a circular footprint with a radius of 0.2m
and move with a maximum speed of 0.5m/s. Robots are initially located on a
circle (equally spaced) with a radius of 10m and the goal locations are set to the
antipodal positions, i.e., each robot’s shortest path is through the center of the
circle. Figure 5 shows example trajectories for 6 robots. The goal is assumed to
be reached when the robot’s center is within a 0.01m radius of the goal location.
The performance of robot iis the negative value of its time of arrival, denoted
as −Ti. Heuristic payoff tables are computed for n= 12 robots, leading to 91
rows. Payoffs for each discrete distribution Niare averaged over (a maximum
of) 20 random permutations on the initial positions of robots.1
1Some discrete distributions, e.g., all robots of one type, do not allow for 20 different
permutations.
Evolutionary Advantage of Reciprocity in Collision Avoidance 9
Fig. 5. Trajectories of 6 RVO robots initially positioned on a circle with goal locations
set to the antipodal positions.
HRVO RVO
VO
Fig. 6. Evolutionary dynamics of a population mixing between the avoidance strate-
gies: velocity obstacle (VO), reciprocal velocity obstacle (RVO) and hybrid-velocity
obstacle (HRVO). Asymptotically stable attractors are depicted by solid circles; unsta-
ble rest points are shown as open circles.
4.3 Results and Discussion
Figure 6 shows the evolutionary dynamics of a population with robots of types
VO, RVO, and HRVO using no truncation of the velocity obstacles, i.e., τ=
∞. All “pure” population states are asymptotically stable fixed points under
the replicator dynamics. However, the strategy space is not partitioned equally
between all attractors, the basin of attraction for RVO is considerably smaller.
Between each pair of strategies, there is one repeller at the uniform mixture. In
addition, there is one saddle point at (0.29,0.49,0.22).
We do not see any dominant strategy for a heterogeneous setting including all
three variations of the velocity obstacle. Also for pairwise comparison between
two strategies, along the faces of the simplex, no strategy is inferior. All three
strategies are evolutionarily stable.
10 Daniel Hennes, Daniel Claes and Karl Tuyls
HRVO RVO
VO
(a) Truncated VO, RVO and HRVO with
τ= 10.
RVO (τ = 2) RVO (τ =10)
RVO (τ = ∞)
(b) Truncated RVO.
Fig. 7. Evolutionary dynamics of collision avoidance with truncation. Asymptotically
stable attractors are depicted by solid circles; unstable rest points are shown as open
circles.
Figure 7(a) shows the dynamics for the same strategies with truncation τ=
10. Strategies VO and HRVO are still asymptotically stable, while RVO is a
repeller. The interior stable fixed point at (0.23,0.49,0.28) has the largest basin
of attraction amounting to more than half of the strategy space.
Introducing truncation leads to significantly different and more complex dy-
namics. In a pairwise comparison (faces of the simplex), RVO is dominated by
VO as well as HRVO. However, the reciprocal velocity obstacle is most robust in
the presence of all three strategies (interior of the simplex). Considering Figure 2,
we see that RVO is the most “aggressive” or least restricting velocity obstacle.
The collision space of a RVO is always a subset of the corresponding VO for
moving obstacles. This is due to the assumption that other robots take care of
half of the collision avoidance. VO and HRVO are both more conservative and
thus restrict the admissible velocity space more.
Finally, Figure 7(b) shows a comparison of different levels of truncation for
the reciprocal velocity obstacle. In particular, we use τ=∞(no truncation),
τ= 2, and τ= 10. In this comparison robots using RVOs with no truncation (τ=
∞) are strictly dominated; both “pure” population states using truncation are
asymptotically stable. Truncation with τ= 2 has the largest basin of attraction.
Truncation with low values for τis less restrictive and robots continue on
a straight path until the truncated velocity obstacle takes affect. As such, the
average time of arrival is shorter and the performance increases. In the pres-
ence of robots employing velocity obstacles with less truncation, distinct use of
truncation leads to situations where robots are “trapped” near the center. In
particular, robots with τ= 2 drive straight towards the center, while robots
with τ= 10 see affect of the velocity obstacles sooner and enter in a spiraling
motion.
Evolutionary Advantage of Reciprocity in Collision Avoidance 11
5 Conclusions and Future Work
We have studied three variations of the velocity obstacle approach in competi-
tion, i.e., the velocity obstacle, the reciprocal velocity obstacle and the hybrid
velocity obstacle. Without truncation all three types perform equally well and
we do not find a dominant strategy. With the use of truncated velocity obstacles
the dynamics become more complex; the reciprocal velocity obstacle is most ro-
bust in the heterogenous system, however, in pairwise comparison this strategy
is inferior.
Our evaluation is based on a scenario commonly used in literature to show-
case the velocity obstacle approach, i.e., robots are initially located on a circle
with their goal locations set to the antipodal positions [15, 13, 7, 1]. A natural
extension is to consider various other scenarios, e.g., robots moving in a free
space or in the presence of obstacles with randomly generated goal locations.
However, it must be taken into account that such a setting requires a global
navigation strategy, which might have an effect on the performance of the local
collision avoidance.
Furthermore, the extension to a less symmetric and stylized configuration
also allows to evaluate aggressive ”straight” driving robots that do not adhere
to any of the collision avoidance obstacles as suggested in the game of chicken.
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