Towards Cooperative Motion Planning
for Automated Vehicles in Mixed Traffic
Maximilian Naumann1 and Christoph Stiller1,2
Abstract— While motion planning techniques for au-
tomated vehicles in a reactive and anticipatory manner
are already widely presented, approaches to cooperative
motion planning are still remaining. In this paper,
we present an approach to enhance common motion
planning algorithms, that allows for cooperation with
human-driven vehicles. Unlike previous approaches, we
integrate the prediction of other traffic participants into
the motion planning, such that the influence of the
ego vehicle’s behavior on the other traffic participants
can be taken into account. For this purpose, a new
cost functional is presented, containing the cost for
all relevant traffic participants in the scene. Finally,
we propose a path-velocity-decomposing sampling-based
implementation of our approach for selected scenarios,
which is evaluated in a simulation.
I. INTRODUCTION
In the field of intelligent vehicles, tremendous progress
has been achieved in the last decades [1]. With the first
successful experiments of close-to-production cars in real
traffic [2], automated driving has gained more and more
attention in public.
In order to improve the reliability and thus the safety
of automated vehicles, but also to increase their effi-
ciency, cooperation is focused on in recent research. Here,
cooperation through explicit communication of (fused)
sensor information and desired driving behaviour [3] as
well as negotiation of possible solutions [4], [5], [6] or
centralized approaches [7] are frequently addressed.
However, as reported in [8], cooperative behavior does
not require V2X-communication. Furthermore, as auto-
mated vehicles will share the road with human-driven
cars at least at the beginning, cooperation with human
drivers in non-V2X-equipped cars is essential. Also, a
natural, cooperative, human-like behavior of automated
vehicles potentially increases their social acceptance.
To the best of the authors’ knowledge, previous motion
planning approaches treated other traffic participants
as obstacles which are to be avoided, similar to static
obstacles like parked cars [2], [9]. While such approaches
can deal with many everyday situations, such as driv-
ing autonomously or following other vehicles, some ma-
neuvers, such as overtaking with oncoming traffic or
*We gratefully acknowledge support of this work by the Tech Center
a-drive
1The
authors
are
with
FZI
Research
Center
for
Information
Technology, Mobile Perception Systems, 76131 Karlsruhe, Germany
naumann@fzi.de
2The author is also with Karlsruhe Institute of Technology (KIT),
Institute of Measurement and Control, 76131 Karlsruhe, Germany
stiller@kit.edu
(a) without signposted right of way
(b) with signposted right of way
Fig. 1: Narrowing, with and without signposted right of way.
passing a narrowing (cf. Figure 1), require combinato-
rial approaches, as already reported by [2]. Still, even
with combinatorial considerations as proposed by [10],
[11], cooperative behavior cannot be implemented: If the
motion prediction of other traffic participants is done
isolated from the motion planning for the ego-vehicle,
the behavior can be foresighted, but not cooperative in
a bidirectional manner [8]. According to a study about
German road traffic, cooperative behavior on average
only occurs in the scale of one cooperative action per
hour per traffic participant [12]. Thus, their treatment
by a separate method, besides the conventional motion
planning, is reasonable.
This paper addresses the problem of cooperative mo-
tion planning without V2X-communication. We propose a
cost functional for trajectory ensembles, consisting of one
trajectory per vehicle. Thereby, we acknowledge the fact
that not only the behavior of other traffic participants
affects us, but also our behavior affects the others in a
closed loop. We consider the motion planning problem as
the problem to find a globally optimal solution for a spe-
cific situation, knowing that every traffic participant has
a different viewpoint considering optimality. The costs
depend on vehicle dynamics, passenger comfort, driving
intention and trajectory clearance, as well as the traffic
regulations, as further outlined in Section II. In this
approach, the prediction of other traffic participants is
integrated into the motion planning. As the assumption
of cooperative behavior might be violated by some traffic
participants, this risk is assessed and the trajectory is
only driven if a safe "plan B" [13] trajectory is still pos-
sible in case of unexpected behavior. An implementation
of this approach is presented in Section III. The proposed
algorithm is finally evaluated in Section IV.
arXiv:1708.06962v1  [cs.RO]  23 Aug 2017
II. GLOBAL OPTIMUM APPROACH
This section introduces the main building blocks of
our approach to cooperative motion planning. Central
to this approach is the assumption that all traffic par-
ticipants are aware of each other and therefore react
on each other’s behavior in a closed loop. Subsequently,
the trajectories for all relevant traffic participants are
considered as one trajectory ensemble, and the quality
of the solution depends on the trajectory of every partic-
ipant separately as well as on the pairwise relation of
the trajectories among each other.
This section is structured as follows: First, the repre-
sentation of one trajectory in the ensemble is introduced.
Subsequently, the cost functional is introduced. Next,
before a solution is selected, the limitations to this
approach are treated by a "plan B".
A. Behavior Policy
Cooperative motion planning is aware of the inter-
action of traffic participants. Therefore, wrong assump-
tions concerning the behavior of other traffic participants
might cause undesired behavior. Even though, theoret-
ically, any feasible behavior is possible, the authors
make the following assumption: Every traffic participant
follows the traffic regulations, as long as this compliant
behavior is physically feasible.
Consequently, assuming perfect perception, a collision
involving our vehicle can only be caused by violating the
traffic regulation without foreseeable reason while our
reaction at the time of violation is insufficient to avoid
the collision.
Arising from this assumption, we pursue the following
policies:
• If we have to give way, we can exclude a collision
independent of others’ behavior.
• If we have the right of way or the situation is not
clearly regulated, we can exclude a collision if others
behave rule compliant.
B. Trajectory Representation
For the representation of a single, deterministic tra-
jectory, the established method of [14] is chosen: The
trajectory x(t) = (x(t), y(t))T is a mapping R →R2, with
tangent angle ψ and curvature κ. A trajectory ensemble
consists of one trajectory per traffic participant: X =
(x1,x2,...), where the superscript describes the partici-
pants identifier.
C. Cost Functional
As proposed in [8], the quality of a solution, given by a
trajectory ensemble, is determined by a cost functional.
The lowest costs denote the best solution. Costs exceed-
ing a certain value represent an infeasible solution. The
cost functional is the sum of the costs of every traffic
participant i
Gtotal =
X
i
Gi.
The costs Gi pursue two main goals: They ensure the
feasibility of the trajectory but also rate its comfort and
effectiveness for a single car. For this reason, the prop-
erties of the trajectory, such as velocity and acceleration,
are rated with multiple evaluation functionals:
The feasibility costs exceed a certain bound if a tra-
jectory is physically not feasible. The pleasantness costs
reflect the wish of the passenger to travel steady and
comfortable, including the perceived safety of the jour-
ney. Furthermore, the costs should motivate compliance
with the traffic regulations. Not yielding is avoided by
upscaling the costs of the vehicle that has the right of
way in the pairwise trajectory costs.
In this approach, the ability to cooperate is associated
with the ability to estimate the cost or quality of a
solution for other traffic participants.
With the above information, the costs Gi per partic-
ipant can be split into costs Gi,0 that only concern the
own trajectory and costs Gi,j that consider the relation
to other trajectories:
Gi = Gi,0 +
X
j
Gi,j
1) Formulation of the trajectory properties: Analog to
[14] the properties of the trajectory that are examined
by the evaluation functionals are
• the velocity v(t) = ˙x(t)
• the acceleration a(t) = ¨x(t)
• the jerk j(t) = ...x(t)
• the distance to the left and right driving
corridor bound dleft(x(t)) and dright(x(t))
• the yaw rate ω(t) = ˙ψ(t) and
• the curvature κ(t) .
Additionally, properties of trajectory pairs describe
their distance to each other. The shortest spatial distance
is described by
dmin(x1(t),x2(t)) = min
t
¡
d(x1(t),x2(t),t)
¢
,
where d denotes a distance measure between states of
different vehicles.
To account for the perceived safety, but also to obey the
traffic regulations, another property is introduced. Here,
we can make use of time-referenced measures, as they
equal a velocity-referenced spatial distance measure. In
general, a collision is only possible if paths overlap. When
determining the criticality, respectively the collision risk,
of two trajectories, their closest point in time and space
is crucial. Regarding a violation of the right of way,
Cooper investigated the post encroachment time (PET) for
specific scenarios [15]. Based on the latter, also regarding
the potential collision zone, we propose the time of zone
clearance (TZC) as a measure for the criticality of two
trajectories with overlapping paths: The TZC is the time
that elapses between the first vehicle leaving potential
collision zone and the second vehicle entering this area,
independent of the right of way (cf. Figure 2).
(a) blue vehicle drove first
(b) black vehicle drove first
Fig. 2: The TZC is the time that the second vehicle takes
to enter the red potential collision zone, assuming constant
velocity.
Given the paths are overlapping and given the trajec-
tories are not colliding, the TZC is calculated as follows:
TZC
=
TZC(xfirst(t),xsecond(t))
=
gap along path
velocity of the second vehicle
=
ssecond(tsecond,in)−ssecond(tfirst,out)
vsecond(tfirst,out)
with ssecond being the path of the vehicle that passes
the collision zone second, vsecond being the scalar velocity
along this path, tfirst,out being the time at which the first
vehicle clears the collision zone and tsecond,in being the
time at which the second vehicle enters the collision zone.
Constant velocity is chosen as passengers cannot foresee
the planned trajectory and as it reflects possible actions
(maximum deceleration or acceleration) best.
If the paths do not overlap, the TZC is defined to be
infinite, if the trajectories collide, it is less or equal zero.
2) Formulation of the evaluation functionals: As in
this work the costs are also calculated for human-driven
cars in order to predict their behavioral decisions, they
should reflect humans’ understanding of the quality of
a trajectory. Therefore, the previously introduced scalar
trajectory properties
f (X) are investigated. Vectorial
properties, such as the acceleration, are therefore split
into their longitudinal and lateral part, using a motion
model.
The costs of a trajectory are subdivided into three
zones:
• comfort zone Zcomf
• discomfort zone Zdisc
• infeasibility zone Zinf
each for positive (+) and negative (−) deviation from the
optimum fopt. The functionals G(f ) expressing the costs
induced by a trajectory property f are called evaluation
functionals.
For the sake of steadiness and piecewise differentia-
bility, all costs are starting from zero at their lower
bound but do not vanish at the start of the next zone.
Accordingly, the total costs G are defined as
G(f ) =





Gcomf
, f ∈Zcomf
Gcomf +Gdisc
, f ∈Zdisc
Gcomf +Gdisc +Ginf
, f ∈Zinf.
The comfort component induces only little costs
G+
comf (f ) = a+ ·
¡
∆f +
comfort
¢2
depending on the distance of f to the optimal value
∆f +
comf = ∆f +
comf(X) = f (X)−fopt.
Consequently, given a comfort threshold Tcomf and as-
suming a comfortable deviation ∆f +
cmargin, the parameter
a+ is to be set to
a+ =
Tcomf
³
∆f +
cmargin
´2 .
The costs G−
comf for comfortable negative deviation are
calculated correspondingly with the parameter a−.
The discomfort costs rise quadratic, but direction-
dependent:
G+
disc (f ) = b+ ·
¡
∆f +
disc
¢2
depending on the distance of f to the upper start of the
discomfort zone f +
disc,start
∆f +
disc = ∆f +
disc(X) = f (X)−f +
disc.
For logical reasons, the parameter b+ should be notably
higher than a+. Negative deviations are treated corre-
spondingly with the parameter b−.
Before the property represents the infeasibility of a
trajectory, the infeasibility costs rise exponentially
G+
inf (f ) = c+ ·
¡
∆f +
inf
¢2 · e|∆f +
inf|
depending on the distance of f to the upper infeasible
value f +
inf minus a margin f +
margin from which the costs
start rising
∆f +
inf = ∆f +
inf(X) = f (X)−
³
f +
inf −∆f +
margin
´
.
Consequently, given an infeasibility threshold Tinf and
assuming a margin ∆f +
imargin, the parameter c+ is to be
set to
c+ =
Tinf
³
∆f +
imargin
´2
· e|∆f +
imargin| .
Further, the infeasibility zone Zinf includes the margin
in this notation. Again, negative deviations are treated
correspondingly with the parameter c−.
Gcomfort
Gdiscomfort
Ginfeasible
G
fopt
f +
disc, start
f +
inf, start
Fig. 3: Composition of the cost function G for a single trajectory
property f : Very low costs around the optimum value fopt,
increasing rapidly in close vicinity of finf.
3) Formulation of the cost functional: With the evalua-
tion functionals, the cost functional for a single property
f is composed as follows (cf. Figure 3):
G (f ) =
G+
comf ·σ(∆f +
comf)+G−
comf ·σ(−∆f −
comf)
+
G+
disc ·σ(∆f +
disc)+G−
disc ·σ(−∆f −
disc)
+
G+
inf ·σ(∆f +
inf)+G−
inf ·σ(−∆f −
inf),
where σ denotes the step function. The right of way of
i over j is acknowledged by adding the comfort-related
costs of vehicle i, upscaled with factor u, to the pairwise
trajectory costs, if i has the right of way:
Gi,j,row = u
¡
Gi,0,comf +Gi,0,disc
¢
.
A suitable choice of u ensures that the right of way is
heeded, but its violation is still feasible, as stated in
Section II-A.
The full cost functional is composed as follows:
Gtotal(X) =
X
i
Ã
Gi,0(xi)+
X
j
Gi,j(xi,xj)
!
with singleton trajectory costs for vehicle i
Gi,0(xi) = Gv +Ga +Gj +Gω +Gκ +Goffset
and pairwise trajectory costs for vehicle i due to vehicle j
Gi,j(xi,xj) = GTZC +Gdmin +Grow.
D. Plan B
In order to obey our previously introduced policy,
plan B trajectories are to be checked, as proposed in
[13]. By doing so, we avoid maneuvering into situations
that lead to collisions, if we made wrong assumptions
concerning the behavior of other traffic participants. As
their execution is unlikely, we accept discomfortable but
feasible trajectories. This corresponds to a neglection of
the comfort terms in the upper cost functional. As with
the previous trajectories, plan B trajectories can be cal-
culated via a local continuous method [14], a sampling-
based method such as RRT∗[16] or other approaches.
E. Selection of Solution
As for passenger comfort, the evaluation of the TZC
should already cause high discomfort costs at around
2 s, a security margin is induced intrinsically by this
approach. Thus, even a very small optimum, represented
by a small range of minimal costs, does not equal a
physically optimal trajectory, that would pass objects
as close as possible in space-time. Rather, it already
contains those security margins that are considered
comfortable by humans and that consequently should be
feasible with measurement uncertainties in the range of
human perception errors. Hence, the optimum point can
be chosen independent of its wideness, as long as a valid
plan B protects the approach against consequences of
wrong assumptions.
III. IMPLEMENTATION
In the following, a first approach for cooperative mo-
tion planning in specific situations, based on the previ-
ously introduced cost functional, is presented.
A. Path-Velocity Decomposition
Several potentially cooperative situations have highly
constraint driving corridors for the traffic participants,
independent of the order and number of traffic partici-
pants. Consequently, we make use of the path-velocity
decomposition (PVD), as introduced by [17]. The calcu-
lation of paths in static environments has already been
widely investigated. Hence, valid paths are considered
predefined (cf. Figure 4) and the implementation focuses
on the velocity profiles along the paths.
B. Sampling
As the optimization problem is non-convex, but the
control variable for the velocity of each vehicle is only
one-dimensional, a classical sampling approach is cho-
sen. Therefore, the trajectories x(t) are approximated by
discretization in equidistant time steps:
xi = x(ti),
ti = t0 + i∆t.
For each car, multiple trajectories are sampled: Start-
ing with an initial position and velocity, a random jerk
sequence determines the velocity profiles and thus the
trajectory. Next, the overall costs of each trajectory en-
semble are calculated. For the solutions with the lowest
costs, the plan B trajectory is checked until a valid plan B
is found.
C. Plan B
Instead of using a different planning method with the
assumption or classical prediction of disadvantageous
behavior of others, we again make use of the PVD: Given
the paths, a collision is only possible in particular areas
that can be determined a priori. Thus, unlike in [13],
no trajectory has to be planned. Rather, the plan B-
consideration can be seen as a “what could I do if”-
consideration. The key questions are: In every time step,
what could the other vehicle do that leads to a collision
with us? And what could we do to avoid this? This
consideration can be split into the following cases:
1) Other vehicle drives first: If the other vehicle drives
first, it can only cause a collision by deceleration. In
reaction, we can decelerate as well. If we can manage
to stop before the collision zone, we have a valid plan B.
2) Ego vehicle drives first: If the ego vehicle drives
first, the other vehicle can only cause a collision by
acceleration. In reaction, we can also accelerate, to still
drive first, or decelerate to stop before the other vehicle
collides with us. As the path is regarded as predefined,
changing the path is not considered.
D. Implications on the cost functional
Since in this implementation, trajectories are dis-
cretized in time, derivatives are approximated by finite
differences. Thus, the functionals of section II-C turn
into functions. Consequently, trajectory properties that
depend on a single minimum, such as TZC, can be largely
affected if this minimum is not sampled. In order to
avoid this, either the sampling rate must be sufficiently
high, or the point of the exact minimum has to be
interpolated. As this implementation is not based on
linear optimization but on sampling, we interpolate the
crucial points.
The jerk is not considered to avoid high order deriva-
tives. Also, the curvature itself is not considered as
the predefined path guarantees the compliance with the
steering geometry. However, it is used to calculate the
lateral acceleration values. Furthermore, the shortest
spatial distance dmin either lies in the collision zone and
is considered by the TZC, or it is not relevant. Hence, it
is neglected as well.
E. Selection of Solutions
As explained in section II-E, criticality protection is
ensured via the costs of the TZC and the check for a
plan B. Consequently, the solution with the lowest costs
and a valid plan B is selected and its ego trajectory is
executed, as long as its costs do not exceed the feasibility-
threshold. In case no solution has a valid plan B, an
emergency braking maneuver is triggered. Note: In in-
car applications, the parallel running classical, reactive
motion-planner would have to take over control in this
case.
IV. EVALUATION
In this section, the method outlined in section III is
evaluated for two scenarios, a left turn at a T-intersection
and passing through a narrowing of the road (cf. Figures
1 and 4).
(a)
(b)
Fig. 4: Left turn at T-junction, with and without signposted
right of way and predefined paths.
A. Simulation
For both scenarios, each with and without signposted
right of way, but sharing the same paths, velocity profiles
were sampled. From the resulting trajectories, ensembles
with one trajectory per vehicle were generated. In order
to reduce computational cost, trajectories that did not
reach the end of the collision zone were excluded from
the cost calculation. Furthermore, colliding trajectory
ensembles were excluded. The remaining ensembles were
analyzed with respect to
• comfort costs
• discomfort costs
• infeasibility costs
• traffic regulation costs.
B. Analysis
As depicted in Figure 5 and 6, the initial states were
chosen in a way that the optima of both vehicles overlap
in the collision zone. In the T-junction scenario, the right
of way is regulated with and without traffic signs. A
violation of the right of way causes high costs so the
optimal solution is following the rules. The trajectory of
the vehicle that has right of way is not interfered (cf.
Figure 5 (2) and (3)).
In the narrowing scenario, the right of way is not
regulated without traffic signs. Here, due to equal cost
parameters, the vehicle that is closer to the narrowing
passes first. Still, traffic signs can overrule this globally
most comfortable solution and shift the optimum (cf.
Figure 1 and 6 (3)).
If a collision can only be avoided by one of the vehicles,
as the other is too close to the collision zone, the optimal
solution is the collision avoidance. Even though this
violates the traffic regulations, the infeasibility costs
overrule discomfort costs and traffic regulation costs.
Further, if we do not interfere a vehicle that has the
right of way, its costs Grow are constantly high, but not
raised by our behavior. In this case, the optimal solution
is that we pass first, without violation of traffic rules.
V. CONCLUSIONS AND FUTURE WORK
In this paper, we presented a new approach to coop-
erative motion planning, able to cooperate with human
Fig. 5: Minimum cost trajectories in the T-junction scenario
with the collision zone marked in grey: (1) for each vehicle
solely on the road, (2) when upper vehicle has right of way
(Fig. 4a), (3) when lower vehicle has right of way (Fig. 4b).
Fig. 6: Minimum cost trajectories in the narrowing scenario
with the collision zone marked in grey: (1) for each vehicle
solely on the road, (2) when no right of way predefined (Fig. 1a),
(3) when left vehicle has right of way (Fig. 1b).
drivers and automated vehicles without requiring V2X-
communication. While the approach is valid for two-
dimensional motion planning, our first implementation
covers several scenarios deploying PVD.
The preliminary results for the simulated scenarios
demonstrate that the method produces safe and com-
fortable cooperative trajectories in a narrowing and a
typical intersection scenario. Individual trajectory costs
have been extended by costs accounting for mutual
comfort and safety of any pair of trajectories. Other
traffic participants have been taken into account by
incorporating their individual costs. The total trajectory
costs for each participant have been segmented into three
areas representing comfortable driving, uncomfortable
driving and collision/infeasibility.
Future work includes real time implementation and
on-road experiments with our vehicle "BerthaOne". Sev-
eral parametrizations will be used for the cost functional,
considering different vehicle types and driver behaviors.
Furthermore, probabilistic trajectories will be accommo-
dated to account for inherent uncertainties in perception
and behavior.
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