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2018 International Conference on Intelligent Systems (IS)
Distributed Cooperative Path-Planning for
Autonomous Vehicles Integrating Human Driver
Trajectories
´
Icaro Bezerra Viana
Centre for Electronic Warfare, Information and Cyber
Cranfield University
Shrivenham, Swindon, United Kingdom
icaro.bezerra-viana@cranfield.ac.uk
Nabil Aouf
Centre for Electronic Warfare, Information and Cyber
Cranfield University
Shrivenham, Swindon, United Kingdom
n.aouf@cranfield.ac.uk
Abstract—This paper considers the problem of cooperative
optimal trajectory generation for autonomous driving involving
a human driver vehicle (HDV) into the traffic scenario. A
distributed model predictive control (MPC) approach to solve
the cooperative path-planning problem is proposed. The novelty
of this paper are in to enhance cooperative path-planning
algorithms, that allows for cooperation between autonomous and
human-driven vehicles. We integrate the trajectory prediction of
the human driver model (HDM) into the framework, such that
the human vehicle influence the behaviour of the other agents.
Mixed-integer quadratic programming (MIQP) it is appealing to
solve the cooperative planning. The proposed method is evaluated
by computational simulations, which show the effectiveness of our
approach for a cooperative overtaking scenario comprising three
vehicles in a three-lane, one way road.
Keywords—autonomous driving, path planning, model predictive
control, collision avoidance
I. INTRODUCTION
Recently, a plenty of works on autonomous driving have
been appearing in the control community. Autonomous driving
systems can help decrease fatalities caused by traffic accidents
[1]. However, this technology has a good deal of aspects to be
improved so as planning obstacle-free paths for a vehicle to
increase highway safety. About this issue, enormous progress
has been made, see the recent reference [2] that proposes a new
formulation of the path planning problem as a mixed-integer
quadratic program (MIQP), applied in challenging scenarios
for the autonomous driving environment.
A challenging research task for autonomous driving is the
coordination of several autonomous cars. This new technology
is the guarantee of making car traffic safer in the future.
Cooperative systems have just recently started to be explored
in the context of automated driving, mainly for the purpose of
exchanging position and sensor data [3].
Various papers have already addressed the cooperative path-
planning problem [4],[5],[6]. Schouwenaars et al. [4] used
an approach to optimal path planning of multiple vehicles
based on mixed-integer programming. Frese and Beyerer [5]
This work has been supported by Innovate UK within the scope of the
project grant MuCCA “Multi-Car Collision Avoidance”.
compared several cooperative path planning algorithms with
respect to computing times and solution quality of the planner
considering dangerous traffic situations. This work considered
a tree search algorithm, the elastic band method, mixed-
integer linear programming, and a priority-based approach.
Wang et al. [6] proposed a cooperative lane change strategy
based on model predictive control in order to attenuate the
adverse impacts of lane change on traffic flow. None of
the aforementioned papers explored an extending cooperative
path-planning approach to traffic where human drivers are
present in on-road scenarios. In the not-too-distant future, it is
feasible that cooperative solutions to driving manoeuvres that
would enable autonomous vehicles to simultaneously coexist
on the roads with vehicles driven by human beings [7].
In this paper, the main contribution is the development of a
cooperative path-planning architecture based on the theoretical
background of distributed MPC extending the scheme to in-
corporate the predicted trajectories of human driving vehicles.
The resulting predicted driver behavior need to be integrated
with the autonomous planning in order to safely interact with
the autonomous vehicles on the road.
The rest of the text is organized as follows: Section II
presents the vehicle model and the hierarchical cooperative
path-planning scheme with its components. Section III de-
scribes the problem solution through model predictive control
using MIQP. Section IV describes the evaluation based on
computer simulations for a three-vehicle overtaking scenario
and Section V contains the conclusions and suggestions for
future works.
II. VEHICLE MODEL AND PROBLEM STATEMENT
The paper considers the following notation: Figure 1 shows
the block diagram with the cooperative planning framework
composed of a MPC-based path-planning. Given a set of
vehicles V{1,2, ..., Nv}, a path-planner for each vehicle
is capable of generate only the two-dimensional trajectories
r(i)
p[r(i)
p,x r(i)
p,y]T∈R2of an vehicle i∈Vto follow a
pre-defined route (i.e., a sequence of roads). To generate the
collision-free reference trajectories, we assume that at every
time step k, each vehicle receives the future plans of the others
978-1-5386-7097-2/18/$31.00 ©2018 IEEE 655
agents r(j)
p[r(j)
p,x r(j)
p,y]T∈R2over a finite horizon, where j
denotes the j-th surrounding vehicle.
On the lower level, the proposed hierarchical scheme is
composed of a tracking controller represented by another MPC
without considering the presence of other vehicles and obsta-
cles. This layer has the role of tracking the trajectory (i.e., a set
of way-points) r(i)
pgenerated by the path-planning layer, which
is then the reference trajectory ¯
r(i)
tfor the tracking controller.
The subscripts pand tare used to represent the trajectories for
the path-planner and the controller, respectively. The computed
longitudinal force Fx,r and δfthe steering angle values are
necessary to control the vehicle and are obtained based on
sensor measurements.
Assumption 1. The route is assumed to be known. This
information is available given a map containing the road
topology, a start and a goal position.
Assumption 2. The current state of the vehicle is estimated
based on measurements from the vehicle sensors and are
available for feedback. This assumption can be satisfied by
using a GPS-aided inertial navigation system.
Plans of
other
vehicles
Reference Trajectory
Fx,r, δf
Route
(Navigation System)
Scenario
Specification
Tracking Controller
Vehicle
Path-Planning
Sensor
Measurements
HDV Trajectory /
Obstacle
Information
Fig. 1. Overview of the hierarchical cooperative planning framework.
In the following sections, we will present the vehicle model
and the definition of the cooperative path-planning problem.
A. Vehicle Dynamics
A non-linear continuous time state-space representation that
describes a kinematic bicycle model [8] is
˙
X=f(X, U ),(1)
with,
˙
X=
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎣
vxcos ψ−vysin ψ
vxsin ψ+vycos ψ
ω
1
m(Fx,r −Fy,f sin δf+mvyω)
1
m(Fy,r +Fy,f cos δf+mvxω)
1
Θ(Fy,f lfcos δf−Fy,rlr)
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎦
.(2)
Define the state vector Xand and the control input vector
Uas
X[rxryψv
xvyω]T∈Rn,U[Fx,r δf]T∈Rm.(3)
in which rxand ryare the coordinates of the center of mass in
an inertial frame, vxand vyare the velocities of the vehicle in
a body-fixed coordinate system. Furthermore ψis the heading
angle of the vehicle that changes at a rate of ω. The geometric
parameters lrand lfdescribe the distance from the vehicles’s
center of gravity to the rear and front axle, respectively.
The control vector Uis composed by the longitudinal force
Fx,r at the rear wheel and the steering angle δfof the front
wheel. Fy,f and Fy,r denote the lateral tire forces at the front
and rear wheels, respectively. For a linear tire model, Fy,i is
defined as
Fy,i =Cαiαi,(4)
where i∈{f,r},αiis the side-slip angle and Cαiis the tire
cornering stiffness.
To design the MPC tracking controller, it is necessary a
linear dynamical model, around a equilibrium condition. Ex-
panding the dynamic equations in (2) around the equilibrium
point (Xe,U
e),
˙
Xe+Δ ˙
X=f(Xe,U
e)+ ∂f
∂X ΔX+∂f
∂U ΔU, (5)
the time-invarying linear system matrices is obtained by the
Jacobian matrix,
A∂f
∂X X=Xe
U=Ue
∈Rn×n,B∂f
∂U X=Xe
U=Ue
∈Rn×m.(6)
B. Path-Planning
The layer responsible for generating the optimal trajecto-
ries is based on mixed-integer quadratic programming in a
receding-horizon fashion. Some pertinent definitions to formu-
late the cooperative path-planning problem are given below,
Definition 1. A human driver model (HDM) integrated with
the path-planner is capable to obtain the current predicted
trajectory ˆ
r(h)
pof a human driver vehicle (h). Given an ad-
ditional error boundary Δeon either dimension, the predicted
trajectory of these vehicles results in:
ˆ
r(h)
p(k)[r(h)
p,x +Δ
ex,r
(h)
p,y +Δ
ey]T.(7)
Assumption 3. The predicted trajectories is generated from a
hypothetical dataset. It is assumed that the predicted errors
656
have constant value along the entire trajectory prediction.
As for the error values we tackled the maximum standard
deviation σfrom a validation model using a hypothetical test
data.
In order to consider the vehicle as a point of mass, the
prediction error are enlarged in all sizes with the lateral W
and longitudinal Ldimensions of the vehicle: Δex=σx+L
and Δey=σy+W. As multi-vehicle collision avoidance will
be ensure by integer constraints, a suitable objective function
to be minimized is,
J(X, U )=
N
j=1
[X(k+j|k)−¯
X]TQ[X(k+j|k)−¯
X]
+
M
j=1
[ΔU(k+j−1|k)]TR[ΔU(k+j−1|k)]
(8)
over a prediction horizon of length Nand along a control
horizon of length M.Q∈Rm×mand R∈Rp×psymmetrical
weighting matrices, with Q≥0and R>0.
In Eq.(8) the state vector X[rp,x vp,x rp,y vp,y]T∈
R4contains the position and velocity in both longitudinal
direction xand lateral direction yof the road, while ΔU
is the incremental control input vector. The control vector
U[ap,x ap,y]T∈R2consists of the longitudinal and lateral
acceleration.
Thus, to achieve the collision avoidance between the au-
tonomous vehicles and the HDVs simply add the following
set of linear inequality convex constraints:
∀k∈{1, ..., tf−1}:|r(i)
p,x(k)−ˆr(h)
p,x(k)|≥Δex
OR |r(i)
p,y(k)−ˆr(h)
p,y (k)|≥Δey,(9)
where r(i)
pis the decision variable of the optimization process
that will integrate the MPC path-planner and ˆ
r(h)is the
predicted position of the vehicle (h)provided by the HDM.
The big-M method [9] can be used to rewrite the collision
avoidance constraints in terms of binary variables b(i,h)
p(k),
yielding
∀k∈{1, ..., tf−1}:
r(i)
p,x(k)−ˆr(h)
p,x(k)≥Δex−Mb(i,h)
1(k)
AND,ˆr(h)
p,x(k)−r(i)
p,x(k)≥Δex−Mb(i,h)
2(k)
AND,r
(i)
p,y(k)−ˆr(h)
p,y (k)≥Δey−Mb(i,h)
3(k)
AND,ˆr(h)
p,y (k)−r(i)
p,y(k)≥Δey−Mb(i,h)
4(k)
AND,
4
p=1
b(i,h)
p(k)≤3,
(10)
where M0is a large constant.
A set of binary variables b(i,h)
p(k)∈{0,1}was added
to the problem at each time step k. The binary variables
are additional decision variables in the optimization problem.
They are such that if bp=0, then the p-th constraint in
(10) is activated. However, if bp=1, then that constraint is
relaxed because the variable Mmoves the upper bound beyond
the solution space. The new constraints (10) are linear in the
decision variables.
Remark 1. To achieve obstacle avoidance with static objects,
we incorporate the same constraints set as described in [10].
This approach represents static obstacles by bounding rectan-
gles, which are assumed to be aligned with the centerline of
a road.
Problem 1. The cooperative path-planning problem for au-
tonomous vehicles is generates Nvoptimal trajectories for
r(i)
p, with i∈I{1,2, ..., Nv}, that minimizes (8), while
respecting the collision avoidance constraints in (10).
III. PROBLEM SOLUTION
This section proposes a solution to Problem 1 by the
incorporation of integer constraints into an MPC-based path
planning formulation. This method allows to include multiple
vehicles in cooperation. We consider the MPC approach as
the baseline algorithm to generate the optimal trajectories for
each one of the Nvvehicles. Once calculated the optimal
solution, the set of waypoints is then converted to the ref-
erence trajectory ¯
r(i)
tfor the tracking controller. The proposed
hierarchical scheme optimize each MPC in parallel to improve
global efficiency.
A. State-Space Model
This section presents an MPC formulation for the path-
planning problem. Consider the following assumption.
Assumption 4. The road curvature is sufficiently small. This
allows consider the road as straight and it is appropriate in
major applications like highways roads.
In accordance with Assumption 4, a design model for
the path-planning stage can be immediately obtained by the
following differential equation,
˙rx=vx,˙vx=ax,˙ry=vy,˙vy=ay.(11)
Eq.(11) describes the vehicle dynamics by a double integra-
tor. Accordingly, the discrete-time linear time-invariant state-
space model is taken into account,
x(k+1) = Adx(k)+Bdu(k),(12)
y(k)=Cdx(k)(13)
with
Ad=⎡
⎢
⎢
⎣
1T00
0100
001T
0001
⎤
⎥
⎥
⎦
,Bd=⎡
⎢
⎢
⎣
T2
20
T0
0T2
2
0T
⎤
⎥
⎥
⎦
,
Cd=⎡
⎢
⎢
⎣
1000
0100
0010
0001
⎤
⎥
⎥
⎦
,
657
where Ad∈R4×4,Bd∈R4×2,Cd∈R4×4was obtained
using the ZOH discretization method, with an integration step
T.
B. Prediction Model
The prediction model following the incremental state-space
formulation [11], can be obtained as,
XN=GΔUM+F,(14)
where XN∈R4N×1stacks the predicted values of the state
vector along N,ΔUM∈R2M×1stacks the incremental control
inputs along M,
G
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
˜
C˜
B0
4×2... 04×2
˜
C˜
A˜
B˜
C˜
B... 04×2
.
.
..
.
.....
.
.
˜
C˜
AM−1˜
B˜
C˜
AM−2˜
B... ˜
C˜
B
.
.
..
.
..
.
.
˜
C˜
AN−1˜
B˜
C˜
AN−2˜
B... ˜
C˜
AN−M˜
B
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
with G∈R4N×2Mand,
F⎡
⎢
⎢
⎢
⎢
⎣
˜
C˜
A
˜
C˜
A2
.
.
.
˜
C˜
AN
⎤
⎥
⎥
⎥
⎥
⎦
ξ(k)∈R4N.
The augmented state vector is defined as [11],
ξ(k)Δx(k)
y(k)∈R8.(15)
C. Road Constraints
Consider the following linear convex inequality constraint
for the position of the vehicle:
rmin ≤r(i)
p≤rmax (16)
The above constraints are employed in this work to model a
single straight road. The vectors rmin,rmax ∈R2contain the
minimum and the maximum values of the coordinates of the
road in each direction. Repeating Eq.(16) along the prediction
horizon N, yields,
[rmin]N≤X
N≤[rmax]N,(17)
which can be rewritten in terms of ΔUMusing Eq.(14),
providing,
AΔUMγ(18)
where,
A−G
G∈R4N×2M
and,
γ[rmax]N−F
F−[rmin]N∈R4N.
D. Collision Avoidance Constraints
Consider the following set of inequality convex constraints
for the distances between the autonomous vehicles and the
HDVs in matrix form, according to (9),
r(i)
p−ˆ
r(h)Δe.(19)
The above constraints are employed in this work to avoid
collisions between the vehicles (i)and (h),∀h=i. The
vector Δe∈R2corresponds to the error boundaries Δex,Δey.
Repeating Eq.(19) along the prediction horizon Nyields
XN−[ˆ
r(h)]N[Δe]N,(20)
where [ˆ
r(h)]N∈R2N×1stacks the constant values of the h-th
human driver vehicle position along the prediction horizon N.
Assumption 5. It is assumed that the HDM estimate the future
position [ˆ
r(h)]N.
Replacing the prediction model (14) into Eq.(20), and
rewriting in terms of the optimization vector X∈R(2M+ndN),
with ndrepresenting the number of binary variables, one can
obtain
TX , (21)
where
XΔUMb(i,h)
pT
,
and
T−G −MInd
2N0nd
2N×nd
2N
G0nd
2N×nd
2N−MInd
2N,
where T∈RndN×(2M+ndN)and the vector is expressed
by,
−[Δe]N+F−[ˆ
r(h)]N
−[Δe]N−F+[
ˆ
r(h)]N∈RndN.
To guarantee that at least one constraint of (10) is activated,
we have 0N×2M[1ndN]TX≤[4]N.
E. Path-Planning
In this work, both path-planner and tracking controllers
obtain the optimal control vector U∗(k)by minimizing a cost
function of the form in Eq.(8). Rewritten (8) in a notation
matrix, we have,
min
XJ(k)=1
2XTHX +MTX.(22)
where in the optimization vector X[ΔUMb(i,h)
p]T, the
additional decision variables b(i,h)
p∈{0,1}N, are constrained
to be binary.
The matrix Hand the vector ϑdefine the terms of the cost
function as follows,
658
1
2H=GTQG +R,MT=2(F−[¯
r(i)
p]N)TQG.(23)
where the controlled output weighting matrix is assumed to
be Q=η×I4Nand the control input weighting matrix is
assumed to be R=ρ×I2M.
The optimal control vector U∗(k)computed at the discrete-
time instant kis given by U∗(k)=ΔU∗(k)+U∗(k−1),
where ΔU∗(k)is the first control vector of ΔU∗
M, which in
turn is obtained by solving the optimization problem in (23)
subject to the constraints,
SX ϑ, (24)
The matrix Sand the vector ϑdefine the linear inequality
constraint set on the optimization variables,
SA
T,ϑγ
∈R(nd+4)N,
where S∈R(nd+4)N×(2M+(nd)N).
F. Trajectory Tracking Controller
The tracking controller design using MPC on the lower
layer of the hierarchy is based on the linear model obtained
in Section II-A. This model is used in order to achieve short
computation times required for real-time application.
Control Constraints
The control constraints can be incorporated in the scheme
and are written in terms of U, resulting,
Umin ≤U≤U
max,(25)
where Umin and Umax are the bound limits on the control
signals as,
Umin =[Fx,r δf]T,Umax =[Fx,r δf]T.(26)
Rewritting the constraint (25) in terms of ΔU, one can
obtain [11],
U(k)=[U(k−1)]N+TNΔU(k),(27)
where
TN=⎡
⎢
⎢
⎢
⎣
I202×2··· 02×2
I2I2··· 02×2
.
.
..
.
.··· .
.
.
I2I2··· I2
⎤
⎥
⎥
⎥
⎦
∈R2N×2M.
Replacing Eq.(27) into Eq.(25), it results
[Umin]N≤[U(k−1)]N+TNΔU(k)≤[Umax ]N.(28)
Finally, Eq.(28) can be writing in terms of the optimization
vector ΔUMas
ΞΔUM≤ξ, (29)
with
ΞTN
−TN,ξ=[Umax ]N−[U(k−1)]N
[U(k−1)]N−[Umin]N,
where Ξ∈R4N×2Mand ξ∈R4N.
MPC for Tracking
To obtain the optimal control vector U∗(k)in the tracking
control layer, it is used the same solution as proposed in (23),
but without consider mixed-integer constraints. The optimiza-
tion problem can be formulated incorporating the additional
control constraints, as
min
ΔUJ(k)=1
2ΔUTHΔU+MTΔU,(30)
subject to (29).
IV. COMPUTATIONAL EVALUATION
In order to illustrate the functionality of the proposed
hierarchical path-planning scheme, it is illustrated in Fig.2 a
traffic scenario with three vehicles and one static obstacle.
The autonomous vehicles are named here as MuCCA Electric
Vehicles (MEVs) [12]. The MEVs (vehicles 1 and 3) drive at
a speed of 25 m/s, while the HDV (vehicle 2) prefers a speed
of 10 m/s. The HDM, embedded in each MEV, predicts one
path for the HDV to avoid the static obstacle moving into the
fast lane, instead of the slow lane. The cooperative trajectory
planning is solved applying the planning framework described
in Sec. II.
The numeric simulation uses the Runge-Kutta 4 as the
solver with an integration step of 0.02 s. The parameters of the
vehicle are shown in Table I. Table II presents the parametriza-
tion adopted for the MPC path-planner in the simulations.
Optimization problems in the MIQP formulation are solved by
employing the academic version of the CPLEX IBM package
[13]. Simulation codes are written in MATLAB/Simulink, and
experiments are performed on a desktop computer with Intel
Core i7-6700U CPU clocked at 3.40GHz with 16GB.
TABLE I. VEHICLE PARAMETERS
Variables Values
Vehicle mass, m1800 kg
Yaw inertia Θ3600 kg m2
Half wheelbase 2.4m
Cornering stiffness cf,cr36000 N m/rad
Length L2m
Width W0.8m
The tracking controller can run at a high frequency and
uses a sampling time T = 0.01 s. It is not necessary long
prediction in the tracking task, because it does not need to
consider obstacles. The control constraints are set to: Fx,r =
−10 N, Fx,r =10N, δf=−1° and δf=1°.
Fig. 3 illustrates the obstacle avoidance scenario during on-
road driving. The rectangular obstacle is centered at O=
659
TABLE II. PARAMETERS OF THE MPC PAT H-PLANNER
Variables Values
Prediction horizon: N=40
Control horizon: M=5
Sampling time T=0.05 s
Controlled output weights: η=[110.11]
T
Control input weights: ρ= [20 20]T
Errors in the HDM prediction: σx=0.4,σy=0.2m
Ve h ic le 2
Ve h ic le 3
Obstacle
Vehicle 1
Fig. 2. Schematic of the initial situation in the example scenario.
[210 1]Tm. It is observed in Fig. 3 that the MEV-1 overtakes
by the left lane, because the presence of the MEV-2 in the
lane of the right side. At the same time, Fig. 4 considers the
manoeuvre where the MEV-2 detect a nearby HDV. It is seen
that the MEV-2 overtakes the HDV by the left side after it has
passed the static obstacle.
Fig. 3. Trajectory generated by the path-planning layer of the MEV-1 for
obstacle avoidance.
Fig. 5 shows the longitudinal force signal as calculated by
Fig. 4. Trajectory generated by the path-planning layer of the MEV-2 during
the collision avoidance with the human driver vehicle.
Fig. 5. Profiles of the longitudinal driving force.
the local trajectory-tracking algorithm. Both MEV-1 and MEV-
2 decelerates for a period to allow the overtaking the static
obstacle and the HDV, respectively. In Fig. 6 is shown the
profiles of the front wheel steering angle. At the moment that
the MEVs avoid the obstacle and the HDV, the control signal
exhibits a smooth driving behaviour.
V. CONCLUSIONS
This paper addressed the problem of cooperative path-
planning, able to cooperate with human driving. The problem
was solved using a hierarchical MPC design with mixed-
integer quadratic program (MIQP) at the planning layer. Inte-
ger constraints was incorporated to ensure collision avoidance
660
Fig. 6. Profiles of the front wheel steering angle.
between the autonomous vehicles and the HDVs. To evaluate
the effectiveness of the method, numerical simulations was
performed to evaluate our approach. We argue that this method
is quite appropriate to attain the on-road autonomous driving
environment, as MIQP showed good results to overtaking
scenarios.
One important assumption in this paper was that the pre-
dicted errors of the HDM have constant value along the entire
trajectory prediction. However, it is desirable for real applica-
tion to take into account a distribution of probability of these
errors and considers other possible paths predicted for the
HDV. For a future work, probabilistic trajectories to account
for inherent uncertainties in the human driver behaviour is
being pursued. This work is planned to be implemented in
real vehicles under the scope of the MuCCA project from
Innovate UK [12].
ACKNOWLEDGMENT
The authors acknowledge the support of Innovate UK by
the financial support under the project grant MuCCA, com-
prised by the consortium with Applus IDIADA, Cosworth
Electronics, Transport Systems Catapult, Westfield Sportscars,
and Cranfield University.
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