Priority-based intersection management
with kinodynamic constraints
Jean Gr´egoire⋆
Silv`ere Bonnabel⋆
Arnaud de La Fortelle⋆†
Abstract— We consider the problem of coordinating a col-
lection of robots at an intersection area taking into account
dynamical constraints due to actuator limitations. We adopt
the coordination space approach, which is standard in multiple
robot motion planning. Assuming the priorities between robots
are assigned in advance and the existence of a collision-free
trajectory respecting those priorities, we propose a provably
safe trajectory planner satisfying kinodynamic constraints. The
algorithm is shown to run in real time and to return safe
(collision-free) trajectories. Simulation results on synthetic data
illustrate the benefits of the approach.
I. INTRODUCTION
A. Motivation
Human error is the sole cause in 57% of all road accidents
and is a contributing factor in over 90% [1], [2]. More-
over, traffic congestion motivates the research to improve
intersection traffic flow. Intelligent transportation systems are
expected to tackle both safety and efficiency issues in the
near future. Many systems have been proposed and they have
proved their ability to increase traffic efficiency – particularly
compared to traffic light systems – and to reduce the risk
of road accidents [3], [4], [5], [6], [7], [8]. Furthermore,
more generally, automated conflict management opens new
perspectives to improve railway [9] and air transportation
systems [10] efficiency. In transportation systems, safety is
usually centralized (e.g. air traffic control, rail management
systems) or at least managed locally in a centralized way
(e.g. traffic lights). In the future, we anticipate there will
be locally full information, e.g. through car-to-car commu-
nication being currently standardized. Obviously there will
be non-communicating entities, sometimes delays or sensing
errors, but our aim is to go from a centralized system in full
information down to more reactive schemes, ensuring safety
first.
B. Related work
The standard approach to multi robot motion planning is to
decompose the problem into two parts, as initiated in [11].
As presented in [12], the first one consists of determining
fixed paths along which robots cross the intersection. The
second one consists of computing the velocity profile of each
robot along its path: this is a well-known problem studied
for applications in automated guided vehicles (AGVs) and
robot manipulators.
⋆MINES ParisTech, Center for Robotics, 60 Bd St Michel 75272 Paris
Cedex 06, France
† Inria Paris - Rocquencourt, RITS team, Domaine de Voluceau -
Rocquencourt, B.P. 105 - 78153 Le Chesnay, France
As first introduced in [13], [14], the path-velocity de-
composition enables to introduce an abstract space: the
coordination space. It is a standard approach to robot motion
planning [15], [16], and the motion planning problem in the
real space boils down to finding an optimal trajectory in the
coordination space that is collision-free with respect to an
obstacle region. The coordination space is a n-dimensional
space (where n denotes the number of robots in the intersec-
tion) and the obstacle-region has a cylindrical structure [17].
In [12], we have revisited the notion of priorities to propose
a novel framework for automated intersection management
based on priority assignment. It is a very intuitive notion:
the priority graph indicates the relative order of robots. Our
framework enables to decompose the motion planning prob-
lem problem in the coordination space into a combinatorial
problem: priority assignment, and a continuous problem:
finding an efficient trajectory with assigned priorities.
The ambition of this framework is to enable more robust-
ness and distribution in future automated intersection man-
agement systems. Indeed, existing intersection management
systems such as proposed in [6], [3], [8] plan the complete
trajectories of robots through the intersection and ensuring
safety requires robots to follow precisely the planned trajec-
tory. By contrast, if priorities only are planned, the priority
graph can be conserved even if some unpredictable event
requires a robot to slow down for some time.
It is now clear that the combinatorial problem of assigning
efficient priorities is inherently difficult, as noticed in [18]
and developed in the priority-based framework in [12]. As a
result, we will only consider in the present paper the issue of
planning ”good” trajectories for already assigned priorities.
When the robots can start and stop instantaneously, it is rela-
tively easy to define an optimal trajectory for fixed priorities.
This trajectory is referred to as the left-greedy trajectory [18],
[12]. However, taking into account acceleration (and higher
derivatives) constraints turns the optimization problem into a
”highly non-trivial” problem (as suggested in the conclusion
of the paper [18]). In the present paper, we address the
challenging problem of finding safe trajectories that respect
this type of constraints. In [19], the problem is formulated
as a mixed integer nonlinear programming problem, and the
solution proposed is suitable only for a ”reasonable” and
fixed number of robots. Moreover, priority assignment and
trajectory planning are not decoupled. In the present paper,
we focus on a low complexity solution to the trajectory
planning problem with assigned priorities which is applicable
for a large and potentially varying number of robots.
arXiv:1310.5828v2  [cs.RO]  5 May 2014
C. Contributions
We introduce a theoretical tool: the braking trajectory,
which is a virtual trajectory obtained letting all robots slow-
ing down as much as possible to stop. The key idea of the
paper is to ensure that at every time-step, the (virtual) braking
trajectory is collision-free. With the proposed planner, robots
are maximally aggressive, i.e. always maximize the distance
travelled at every time-step. However, they do not accelerate
if the virtual braking trajectory becomes unsafe or violates a
priority, i.e. they ensure the existence of a failsafe maneuver
for the system of robots at any time. We present a trajectory
planner with assigned priorities that consists of just-in-time
braking and is proved to return collision-free trajectories
respecting the assigned priorities.
Section II and III present the modelling assumptions and
recall the basics of the priority-based framework of [12].
Section IV introduces the motion planner algorithm along
with its safety guarantees. Finally, simulation results of
Section V illustrate the efficiency of the approach.
II. MODELLING ASSUMPTIONS
A. The coordination space
We assume that robots are constrained to follow prede-
fined paths to go through the intersection. The paths are
not necessarily straight lines: robots are just considered as
driving along fixed tracks. This can be achieved by a low-
level controller. This standard assumption [13], [20], [21],
[22], [7] fits well intersections in a road network, where
robots travel along lanes.
Every robot i follows a particular path γi and we denote
xi
∈R its curvilinear coordinate along the path. The
configuration of the system of robots is x = (xi)i∈{1...n} and
we denote x(t) the evolution of x through time t ∈[0, T].
Figure 1 illustrates the path following assumption.
γ1 = γ2 = γ3 = γ4 
γ5 = γ6 
γ7 = γ8
γ11 = γ12
γ9 = γ10
8
10
9
7
12
11
1
2
3
4
5
6
Fig. 1.
The path following assumption. All robots in the same lane
(depicted with the same color) travel along the same geometric path with
independent velocity profiles.
The configuration space χ is known as the coordination
space [23], [17], [13]. In the rest of the paper, {ei}1≤i≤n
denotes the canonical basis of χ. The use of the coordination
space and the results of this paragraph are standard [17].
As every robot occupies a non-empty geometric region,
some configurations must be excluded to avoid collisions
between robots. The obstacle region χobs is the open set
of all collision configurations. χfree = χ\χobs denotes the
obstacle-free space.
A collision occurs when two robots occupy a common
region of space, so that the obstacle region can be described
as the union of n(n−1)/2 open cylinders χobs
ij
corresponding
to as many collision pairs: χobs = ∪i>jχobs
ij . Each cylinder
χobs
ij
is assumed to have an open bounded convex cross-
section (in the plane generated by ei and ej). Figure 2
displays the obstacle region and a collision configuration for
a two-path intersection. We assume χobs ̸= ∅(otherwise,
coordination is not required), so the boundedness condition
ensures that inf χobs and sup χobs both exist.
x1
x2
χobs
Υ1 
Υ2
Fig. 2.
The left drawing depicts two robots in collision in a 2-path-
intersection. The right drawing depicts the corresponding configuration in
the coordination space that belongs to the obstacle region.
B. Kinodynamic constraints
In this paper, we propose to take into account the technical
constraints of the robots at the motion planning phase.
These include kinematic constraints (maximum velocity,
maximum curve radius, etc.) and dynamic constraints (lim-
ited acceleration, adherence, jerk, etc.). Let p denote the
degree of the constraints and n the number of robots. Let
s(t) = (x x′ · · · x(p))(t) ∈Rn×(p+1) denote the state of the
system. We let x(t) = π(s(t)) denote the first column
of the state s(t), that is the position of all robots. We
say a trajectory x respects the kinodynamic constraints C
if: ∀i ∈{1...n}, ∀t ∈[0, T], we have si(t) ∈Ci with
Ci ⊂Rp+1 representing the constraints for robot i and
C = Q
i∈{1...n} Ci ⊂Rn×(p+1).
Note that every robot can have different constraints Ci,
and Ci can not necessarily be expressed in a product form
(for example, the constraint on the velocity can depend on
the position).
We assume that the kinodynamic constraints are such that
the set of reachable positions from a given state in finite time
is bounded. More precisely, the set of reachable positions
from state s0 in a time-length t:
χreach(s0, t) =

x(t)

x respects the constraints C
s(0) = s0

(1)
is assumed to be continuous with respect to s0 and to be
a bounded hypercube of x0 + Rn
+. Note that, the above
assumptions imply in particular that:
1) robots cannot travel backwards in the intersection,
2) and from a given state s0, the set of reachable positions
in finite time is bounded, and the bounds depend on the
state s0 of the robots (position, velocity, acceleration,
etc.).
III. THE PRIORITY-BASED FRAMEWORK
In this section, we recall the basics of priority-based
intersection management introduced in our previous work
[12].
A. The priority graph
Consider the region χobs
i≻j defined as follows and depicted
in Figure 3:
χobs
i≻j = χobs
ij
−R+ei + R+ej
(2)
xi
xi
xj
χobs
xi
xj
χobs
j>i
xi
xj
χobs
i>j
xi>j(t) 
xj
χobs
xj>i(t) 
xi>j(t) 
xj>i(t) 
Fig. 3.
The top drawings represent in the plane (xi, xj) the obstacle
region χobs, a collision-free trajectory xi≻j respecting priority i ≻j
and a collision-free trajectory xj≻i respecting priority j ≻i. The bottom
drawings depict χobs
i≻j and χobs
j≻i.
We define a natural binary relation corresponding to pri-
ority relations between robots. A collision-free trajectory x
induces a binary relation ≻on the set {1...n} as follows. For
i ̸= j s.t. χobs
ij
̸= ∅, i ≻j if x is collision-free with χobs
i≻j.
The priority relation can be described by a graph G
with nodes {1...n}, where each edge represents the relative
priority of a pair of robots. Given a collision-free trajectory
x, the priority graph is defined as the oriented graph G whose
vertices are V (G) = {1...n} and such that there is an edge
from i to j if i ≻j, we write (i, j) ∈E(G) where E(G)
denotes the edge set. An example of a priority graph for 3
robots along 3 distinct paths is described in Figure 4.
B. Problem formulation
The initial state of the robots is sinit, and the goal region
is χgoal =
�sup χobs + Rn
+


⊂χfree. A feasible trajectory
for the considered problem is a trajectory x : [0, T] →χfree
respecting constraints C such that s(0) = sinit and x(T) ∈
3
1
2
3
1
2
3
1
2
3
1
2
Fig. 4.
Two representations of priority relations. Robots along a path in
foreground have priority over robots along a path in background.
χgoal. The multiple robot motion planning problem consists
of finding a feasible trajectory. The benefit of the priority-
based approach is that the priority graph captures the discrete
part of the problem that consists of assigning the relative
order of robots through the intersection.
When the priority graph G is fixed, for all (i, j) ∈E(G),
the trajectory must be collision-free with regards to χobs
i≻j.
Given, a priority graph G, the collision region with regards
to priorities G is merely defined as:
χobs
G
=
[
(i,j)∈E(G)
χobs
i≻j
(3)
It is natural to define χfree
G
= χ \ χobs
G , so that {χfree
G , χobs
G }
form a partition of χ. In this paper, we focus on the problem
on finding a feasible trajectory respecting assigned priorities,
i.e. a trajectory x : [0, T] →χfree
G
respecting constraints C
such that s(0) = sinit and x(T) ∈χgoal.
IV. MOTION PLANNER WITH ASSIGNED PRIORITIES
The key idea is that if robots wait to be at the boundary
of the collision region to brake (as it is the case without
dynamic constraints in [12]), collisions will occur because
robots can not stop instantly. That is why we need to
anticipate the approach of the collision region. This can be
done introducing two virtual trajectories as follows.
A. Introducing maximal and minimal trajectories
The minimal (resp. maximal) trajectory from state s0,
denoted x(s0, t) (resp. x(s0, t)), are defined bellow:
x(s0, t)
=
minχreach(s0, t)
x(s0, t)
=
maxχreach(s0, t)
These are the lower and upper bounds of the hypercube
χreach(s0, t). One can view the minimal trajectory as a brak-
ing trajectory, and the maximal trajectory as an accelerating
trajectory. The concepts are illustrated by Figure 5 where the
kinodynamic constraints have the special following form:
Cacc
i
=

(xi, x′
i, x′′
i )

0 ≤x′
i ≤vmax
i
amin
i
≤x′′
i ≤amax
i

(4)
x'i
x"i
vmax
amax
amin
Ci
xi
t
xi(t)
t0
x(s(t0),t-t0)
x(s(t0),t-t0)
Fig. 5.
The left drawing depicts an example of kinodynamic constraints
where robots have uniform minimal/maximal velocity and acceleration along
their paths. The right drawing depicts the corresponding minimal/maximal
trajectories x(s(t0), t) and x(s(t0), t).
B. The motion planner
The time t is first discretized, and the trajectory of the
robots x(t) is computed iteratively as described in the
following Algorithm 1. Indeed, at every time-step t, the
trajectory up to time t + ∆T can be computed as follows:
• Cycling through all robots, we select a particular robot
i (line 5);
• We compute a virtual path χvirtual (line 12) that would
be followed by the robots if:
– in the next time step, robot i accelerates as much as
possible while all other robots decelerate as much
as possible (lines 6-11);
– afterwards, all robots including i brake as much as
possible (see the second term of the concatenation
at line 12)
• If this virtual path is such that no collision and priorities
are respected, it means there exists a failsafe maneuver
such that robot i accelerates as much as possible in the
next time step, and we let it do so. Otherwise, robot
i must brake (lines 14-18). Thus, at each time-step t,
each robot i exclusively follows its maximal or minimal
trajectory in the next time-step.
The defined trajectory thus appears as a natural extension
of the left-greedy trajectory introduced in [18], in the sense
that in the absence of kinodynamic constraints (p = 1), it
coincides with it. Indeed, in this case the robots can stop
instantly and the block from Line 6 to Line 17 simply
consists of checking that maximum speed during the next
time-step is safe: if it is not the case the robot is stopped.
Note also that if the state s(t) is such that the braking
trajectory from s(t) is collision-free, the state s(t) is not an
”Inevitable Collision State” (ICS), as defined in [24] because
the braking trajectory is collision-free, that is, there exists a
particular collision-free trajectory starting from state s(t).
C. Safety guarantees
The theorem below exhibits the safety guarantee provided
by the proposed motion planner.
Theorem 1 (Safety guarantees). Assume that there exists
some feasible trajectory respecting priorities defined by G
and the initial state sinit is such that the initial braking
Algorithm 1 The motion planner with assigned priorities
Input: sinit, feasible priority graph G
function MAXIMALLYAGRESSIVETRAJECTORY
T ←0
s(0) ←sinit
while x(T) /∈χgoal do
5:
for i ∈{1...n} do
for t ∈[0, ∆T] do
for j ̸= i do
˜sj(t) ←sj(s(T), t)
end for
10:
˜si(t) ←si(s(T), t)
end for
χvirtual ←π(˜s([0, ∆T])) ∪x(˜s(∆T), R+)
if ∃(j, i) ∈E(G) s.t. χvirtual∩χobs
j≻i ̸= ∅then
si(T + ∆T) ←si(s(T), ∆T)
15:
else
si(T + ∆T) ←si(s(T), ∆T)
end if
end for
T ←T + ∆T
20:
end while
return (x(t))t=0···T
end function
trajectory x(sinit, t) is collision-free. Then, for sufficiently
small ∆T, Algorithm 1 terminates and returns a (collision-
free) feasible trajectory respecting priorities G.
Proof. It is assumed that that there exists some feasible
trajectory respecting priorities defined by G, so that G is a
feasible priority graph as defined in [12], and the trajectory
cannot reach a deadlock configuration (for sufficiently small
∆T). Until χgoal is reached, at any time there is at least one
robot i0 at a coordinate lower than sup{xi0 : x ∈χobs}
moving forward. Indeed if this was not true it would mean
that the robots have reached a deadlock configuration. There
is thus a lower bound, say µ, for the distance travelled by
some of the robots in a time-length ∆T, depending on the
constraints. This implies x necessarily reaches χgoal in finite
time (of order at most O(n/µ)) and Algorithm 1 terminates.
Now we prove that, at every time step, the braking trajec-
tory from the current state is collision-free. We begin with a
preliminary useful property, that is a direct consequence of
the definition of χobs
j≻i and is easily seen on Figure 3.
Property 1. Given i, j ∈{1...n} and two configurations
x, y ∈χ satisfying yj ≥xj and yi ≤xi, we have:
x ∈χfree
j≻i ⇒y ∈χfree
j≻i
(5)
The initial braking trajectory x(sinit, t) is assumed to
be collision-free. Now, assume that for some t0 = k∆T,
x(s(t0), t) is collision-free. In the next time step, for any
priority (j, i) ∈E(G), there are two options for the robot
with lower priority:
• either i brakes as much as possible. The fact that
x(s(t0), t) is collision-free with χobs
j≻i implies that
x(s(t0 + ∆T), t) is also collision-free by Property 1,
since we have for all t ≥0:
xi(s(t0 + ∆T), t)
=
xi(s(t0), t + ∆T)
(6)
xj(s(t0 + ∆T), t)
≥
xj(s(t0), t + ∆T)
(7)
• or i accelerates as much as possible; in this case, the
virtual path χvirtual is collision-free with respect to
χobs
j≻i. Then, consider the state ˜s(∆T) defined as:
˜sj(∆T)
=
sj(s(t0), ∆T)
(8)
˜si(∆T)
=
si(s(t0), ∆T)
(9)
Since the virtual path χvirtual is collision-free with
χobs
j≻i, x(˜s(∆T), t) is also collision-free (see the second
term in the concatenation at Line 12). It implies that
x(s(t0 + ∆T), t) is also collision-free by Property 1,
since we have for all t ≥0:
xi(s(t0 + ∆T), t)
=
xi(˜s(∆T), t)
(10)
xj(s(t0 + ∆T), t)
≥
xj(˜s(∆T), t)
(11)
Hence, at every time step, the braking trajectory is
collision-free. Now, we prove that there is no collision
between time steps. Again, at every time step t0 = k∆T,
the are two options:
• either i brakes as much as possible. The fact that
x(s(t0), t) is collision-free with χobs
j≻i implies that for
t ∈[t0, t0 + ∆T], s(t) is also collision-free by Prop-
erty 1, since we have for all t ∈[t0, t0 + ∆T]:
xi(t)
=
xi(s(t0), t)
(12)
xj(t)
≥
xj(s(t0), t)
(13)
• or i accelerates as much as possible; in this case, the
virtual path χvirtual is collision-free with respect to
χobs
j≻i. Then, consider the trajectory ˜x(t) for t ∈[0, ∆T]
defined as:
˜xj(t)
=
xj(s(t0), t)
(14)
˜xi(t)
=
xi(s(t0), t)
(15)
Since χvirtual is collision-free with χobs
j≻i, ˜x(t) is also
collision-free (see the first term in the concatenation at
Line 12). It implies that s(t) is also collision-free for
t ∈[t0, t0 + ∆T] by Property 1, since we have:
si(t)
=
˜si(t −t0)
(16)
sj(t)
≥
˜sj(t −t0)
(17)
As a result, x is collision-free with respect to χobs
G
at every
time-step and reaches χgoal: it is a collision-free feasible
trajectory respecting priorities G.
V. SIMULATIONS
The algorithms presented in this paper have been imple-
mented into a simulator coded in Java. Our algorithms have
proved their ability to run in real-time.
A. Setting and results
Only straight paths are implemented (for simplicity’s sake)
and all robots are supposed to be circle-shaped with a
common radius R. The kinodynamic constraints of the robots
concern only the maximal velocity and minimal/maximal
acceleration. Moreover, all robots are supposed to have
identical kinodynamic constraints.
∀i ∈{1...n}, Cacc
i
=

(xi, x′
i, x′′
i )

0 ≤x′
i ≤vmax
amin ≤x′′
i ≤amax

.
(18)
In the simulation results presented in this section, we
take as priority assignment policy the maximally aggressive
priority assignment policy that consists for every robot of
taking priority over another robot if it reaches the conflicting
region first. This priority assignment policy can lead to
deadlock configurations (see [12]), but with a very small
probability in case of low traffic density as in the presented
simulations. This policy is used for the sake of simplicity, the
priority assignment policy not being the focus of this paper.
Simulations
have
been
carried
out
for
the
4-path-
intersection depicted in Figure 7. At full speed, the distance
travelled in one time-step is R and at full acceleration, 20
time-steps are required for the robots to reach full speed.
Figure 6 depicts the increase in travel time for different
traffic densities. The increase in travel time is the delay due
to coordination, i.e. the difference with the ideal travel time
which is the travel time of robots in the absence of other
robots. It is expressed in percentage of the ideal travel time.
The increase in travel time vanishes as the density approaches
0 since it becomes very unlikely that they need to coordinate
to avoid collisions. The traffic density in percentage is the
ratio between the actual traffic density and the maximum
traffic density (continuous flow of robots). The robots are
generated randomly at a constant rate over time. The video
of the simulation for a traffic density of 10% is available at
http://youtu.be/bJHdf3AbIlI.
10.0%
15.0%
Increase in travel time (%)
0.0%
5.0%
0.0%
2.0%
4.0%
6.0%
8.0%
10.0%
Increase in travel time (%)
Traffic density (%)
Traffic density (%)
Fig. 6.
Simulation results: plot of the averaged increase in travel time
against the traffic density for the intersection of Figure 7
Fig. 7.
The 4-path-intersection used for simulations
B. Comments
First of all, our algorithm succeeds to work in real time,
and one can observe in simulations (notably on the video)
that collisions never occur. This confirms the fact that the
planner guarantees safety under dynamic constraints. One
can see in Figure 6 that at a traffic density of 10% on each
path, the increase in travel time due to coordination to avoid
other robots is less than 15% which seems a low price to
pay to ensure safe coordination. Note that we do not present
simulation results at higher traffic densities because it would
require to define a more complex priority assignment policy
(at least to avoid deadlocks), which is a challenge in itself,
and beyond the scope of the present paper.
VI. CONCLUSIONS AND DISCUSSION
The results presented in this paper prove that when
priorities are assigned, it is possible to plan a safe and
quite efficient trajectory respecting the priority graph and
the dynamic constraints of the robots. The use of the braking
trajectory enables to anticipate the need to brake just-in-time,
and as a byproduct provides robustness guarantees since
there exists a collision-free braking maneuver at any time.
If the robots drift from the planned trajectory but if no
priority has been violated, it is possible to run the motion
planner from a new initial state to get a new feasible
trajectory respecting the assigned priorities. This reflects that
the method proposed in this paper is inherently a feedback
motion planning approach (see [17], chapter 8). We are
currently turning the planning algorithms of this paper into
a feedback control law that aims at coordinating robots with
assigned priorities. The idea is to define a control law gG(s)
that maps every state s to the control to apply in the next
time step. The control law gG is in charge of coordination,
ensuring that collisions are avoided and that priorities G are
respected. Robots do not have to follow precisely a planned
trajectory, they just have to be aware of the priorities and
to respect the control law. The benefit of the approach is
that it ensures safe coordination as long as priorities G
are respected, which is much easier to robustly ensure than
following precisely a planned trajectory. This opens avenues
to build multiple robot coordination systems much more
robust with regards to uncertainty in control and sensing.
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