vehicle trajectory optimization

Vehicle Trajectory Optimization for Application in
Eco-Driving
Felicitas Mensing1,2, Rochdi Trigui1, Eric Bideaux2
1 IFSTTAR Bron - LTE
2 AMPERE - INSA Lyon
Abstract—To reduce fuel consumption in the transportation
sector research focuses mainly on the development of more
efficient drive train technologies and alternative drive train
designs. Another and immidiately applicable way found to reduce
fuel consumption in road vehicles is to change vehicle operation
such that system efficiency is maximized. The concept of Eco-
driving refers to the change of driver behavior in a fuel saving
way or more generally in an energy saving way.
In this paper system efficiency of a vehicle is optimized using
a dynamic programming optimization approach. Given a drive
cycle a so called ’eco-drive cycle’ is identified in which a vehicle
performs the same distance with the same stops in equivalent
time, while consuming less fuel.
I. INTRODUCTION
The transportation sector is a major contributor to air
pollution and consumer of scarce non-renewable fossil fuels
[1]. With the rising fuel prizes and environmental concerns
researchers are currently investigating new technologies to
increase fuel efficiency and reduce emissions for road vehicles.
Alternative drive train designs like the electric and hybrid
vehicles are expected to have high potentials to decrease fuel
consumption due to their ability to regenerate kinetic energy in
the braking phase. In addition, in electric vehicles the electric-
ity used can be generated from alternative, renewable energy
sources. Another approach taken to increase system efficiency
in vehicles is to improve current drive train technologies. By
increasing the efficiency of individual components in the drive
train overall system efficiency is maximized.
A third way to reduce fuel consumption of road vehicles
is to adapt driver behavior such that vehicle operation is
optimized. Here an already existing vehicle configuration is
used such that, for a desired mission, the fuel consumed is
minimized. This concept of adapting driver behavior in a fuel
saving way is often referred to as ’Eco-Driving’.
It is well known that fuel economy in road vehicles does
not only depend on the drive train but also on its operation.
This has first been demonstrated in 1977 by Schwarzkopf [2].
Since then several works have shown that optimizing vehicle
operation for a given mission can result in significant gains
in system efficiency [3], [4], [5], [6]. Another study done
by Ericsson [7] shows that certain driving patterns have an
important effect on emissions and fuel consumption.
Implementing and applying their results in real time driving
has been difficult since an interface between the optimization
and the driver has to be designed. Today several approaches to
develop eco-driving strategies can be found. In most countries
eco-driving courses are offered. But these have shown to
reduce fuel consumption only for a short time period [8].
Other eco-driving tools consist of a display, which gives the
driver advice on the optimal vehicle operation [9], or an active
accelerator pedal, which influences the rate of acceleration of
the driver by applying a resistance on the gas pedal [10].
In this work we suggest a strategy to compute a so called
eco-drive cycle, given a general drive cycle. Using numerical
optimization tools the best operation, which results in the
same trip distance, with equivalent stops, in the same final
time, will be computed for a given drive train configuration.
For verification the vehicle will be simulated in a forward
facing manner over the original drive cycle as well as over
the determined eco-drive cycle. It is expected that the fuel
consumption will be reduced from the original drive cycle to
the eco-drive cycle. In analyzing the results the potentials in
applying the suggested optimization algorithm to eco-driving
will be investigated.
In the following the system considered for optimization is
presented. In Section III the approach suggested in this work
is introduced. The method of optimization is discussed and the
generation of an eco-drive cycle, given a general drive cycle,
is explained. Results of application of the proposed strategy
in simulation will be shown in Section IV.
II. SYSTEM MODEL
The concept suggested in this paper is applicable to any
vehicle architecture: conventional, electric or hybrid drive-
train. For simplicity it will here be demonstrated on the exam-
ple of a conventional drive-train. For optimization purposes a
backward (inverse) vehicle model, where the engine operation
is computed given the operation of the vehicle, is needed.
The vehicle considered is a compact passenger car with a
mass (M) of 1020kg. The vehicle can be modeled as a standard
drive train as seen in the schematic in Figure 1 including the
final drive reduction, a 5-speed gear box, a clutch, auxiliary
losses and the engine. The engine used in this work represents
a 1.5L diesel engine with common rail direct fuel injection.
The vehicle model was constructed using the VEHLIB library,
which was previously developed at IFSTTAR [11].
In this inverse modeling approach the losses and operations
of each component is calculated in the direction opposite to
the power flow. In general the vehicle is propelled by the
engine, the power source is therefore the internal combustion
engine. Given the output of the engine the acceleration of
978-1-61284-247-9/11/$26.00 ©2011 IEEE

Figure 1. Clio 1.5L Drive Train Schematic
the vehicle can be computed. In this approach, however, the
required operation of the engine is back-calculated assuming
a certain speed and acceleration of the vehicle.
The force required at the wheels to satisfy a certain velocity
and acceleration of the vehicle can be calculated in a quasi-
static approach. The force, often referred to as resistance force,
is computed as a sum of the rolling resistance (Froll ),the
aerodynamic drag (Fdrag), the road grade resistance (Fgrade)
and the acceleration force (Fa=Ma):
Fres =Froll +Fdrag +Fgrade +Fa.
With this and the vehicle speed (v) the wheel operation can
be expressed as:
Tw=Fres ∗Rtire
ω
w=v/Rtire
where Rtire represents the radius of the tire.
Given the wheel operation the engine operation can be found
by modeling the losses of the final drive, the gear box, the
clutch and the auxiliaries. In this paper the efficiency of the
final drive reduction and the gears are assumed to be constant
with respect to operating torque and speed. The gear selection
is optimized such that the instantaneous fuel rate is minimized.
Losses in the clutch exist if the engine speed is bellow idle and
the clutch is slipping. The auxiliary components are assumed
to absorb constant 300W. The diesel engine is modeled using
an engine efficiency map with maximum efficiency of 40%.
Knowing the engine speed and torque the instantaneous fuel
consumption can then be calculated using a look up table.
Using this model the instantaneous fuel consumption can
be computed as a function of vehicle velocity (v) and vehicle
acceleration (a):
˙mfuel =f(v,a)
This backward facing model can now be utilized in the
optimization process to find the fuel consumption for a chosen
vehicle speed and acceleration. Once optimal operation was
determined as a velocity trajectory over time the gains of
this eco-driving algorithm was verified with a forward facing
model. Rather than defining the vehicle speed and acceleration
the driver operation of the gas and break pedal is here used
as an input to the simulation. In order to follow the desired
speed profile the driver input was implemented using a PID
controller.
III. METHODOLOGY
In the following the velocity trajectory optimization problem
will be discussed and a method to generate an eco-drive cycle
is presented.
A. Optimization
Optimization of a velocity profile is a well known prob-
lem in literature. Several approaches can be found where
researchers optimize the energy utilization of road vehicles for
a given trip [3], [4], [5], [6]. While most works consider energy
or fuel as the cost function to be optimized, other studies have
been done where time consumed over a trip is minimized [12].
In this study the velocity profile of a vehicle is optimized
such that a given mission is achieved in a fixed time consid-
ering overall fuel consumption as the cost. The system to be
optimized can then be described in a discrete way by
xi+1=xi+viΔt+1
2aiΔt2(III.1)
vi+1=vi+aiΔt(III.2)
where xis the distance, vis the longitudinal vehicle velocity,
ais the longitudinal vehicle acceleration and tthe time. The
system consists of two states, the distance driven (x) and the
velocity of the vehicle (v). The control variable is represented
by the acceleration (a).
The objective function is defined by
J=∑t=tf
t=0Jfuel(t)=∑t=tf t=0˙mfuel(t)
with the following constraints applied to the final and initial
states:
x(0)=x0x(tf)=xf
v(0)=v0v(tf)=vf
tf=T
Commonly used optimization methods to solve the energy
utilization problem for road vehicles are the Pontryagin’s
Maximum Principle [13] and the Dynamic Programming Op-
timization Method [13]. In this work a 3-dimensional dynamic
programming approach, after the example of Hooker [14], was
chosen.
In the dynamic programming optimization the search for the
optimal trajectory is simplified using the Bellman principle
while searching from the final state backward in time. The
Bellman Principle of Optimality states the following [13]:
’An optimal policy has the property that whatever the initial
state and initial decision are, the remaining decisions must
constitute an optimal policy with regard to the state resulting
from the first decision.’ In general the dynamic programming
method has two major parts, as shown in Figure 2. These are
the calculation of optimal cost and indexes to the final state

Figure 2. Dynamic Programming Optimization Flowchart
backwards in time under utilization of the Bellmann Principle
and the computation of the optimal trajectories retracing the
stored indexes forward in time. In the following the method of
dynamic programming optimization in the three dimensional
approach will be described in more detail:
In a 3-dimensional approach three variables are to be
defined at each state, here time (t), distance (x) and velocity
(v):
X=⎡
⎣
x1
x2
x3
⎤
⎦=⎡ ⎣
t
x
v
⎤
⎦
Initially the possible range of time, distance and speed
are discretized and a state Xk,i,jrefers to the state
[t(k)x(i)v(j)]T. The initial and final distance and speed
at t(0)and t(N)are fixed and denoted
Xo=⎡
⎣
to
xo
vo
⎤
⎦and Xf=⎡
⎣
tf
xf
vf
⎤
⎦.
Beginning the search for the optimal trajectory from the
final state the optimal costs J∗
[N−1,i,j]=J[N−1,i,j−>N,if,jf]and
indexes I[N−1,i,j]of the respective trajectories are stored for all
possible states at the last time step N−1. The state transition
diagram to visualize the process can be seen in Figure 3.
With this we can assume that the optimal trajectories for all
possible values of i2and j2at some point in time N−a+1
are known and their costs to reach the desired final state are
given by J∗
[N−a+1,i2,j2]. The optimal trajectory from some state
X[N−a,i1,j1]to the final state can then be found by comparing
the sum of costs of the state transitions between X[N−a,i1,j1]to
X[N−a+1,i2,j2]and the optimal cost from X[N−a+1,i2,j2]to Xffor
all possible [i2,j2]. The cost of the optimal trajectory at N−a
is then given by:
J∗
[N−a,i1,j1]=min
i2,j2
(J[N−a,i1,j1−>N−a+1,i2,j2]+J∗
[N−a+1,i2,j2])
Storing the costs J∗
[N−a,i,j]and indexes I[N−a,i,j]for all a,i,
and jthe optimal cost from Xoto Xfis found after Niterative
steps. In order to find the sequence of states used to result in
this cost the indexes are used to retrace the trajectory forward
in time.
Xf
tf
xf
distance x
time t
velocity v
Xo
J*[N-1,i,j->N,if,jf]
Figure 3. Dynamic programming optimization method
Using dynamic programming the optimal solution is found
by searching all possible proceeding state values at each step
in time. The computational cost of this 3-dimensional method
is very high in comparison with a two dimensional approach.
For example, given a problem with two dimensions and K
possible values at each dimension the method searches at
each of the K possible time steps the K possible state values.
The computational cost becomes K∗KC =K2C, where Cis
the cost to calculate one edge. However, when the problem
consists of three dimensions (as here), and again K possible
values are given for each dimension, the method has to search
each of the K time steps at all possible combinations (K2)
of the other two variables. The resulting computational cost
becomes K∗K2C=K3C. Hence the computation time of a
dynamic programming optimization grows exponentially with
the dimension size. A three dimensional approach with the axis
time, distance and speed is used here because it is necessary
in order to fix and satisfy initial and final conditions in all of
these dimensions. In future work, and in order to implement
the developed concept in real time more rapid optimization
methods should be explored.
As previously stated the system can be described with
two states, therefore using a 3 dimensional approach for
optimization results in a dependency of the three axis. While

Hooker[14] used interpolation to fix the resulting distance, xi,
for a given time, ti, and speed, vi, to the grid it was found
in these studies that the dependency of the three dimensions
allows to make an intelligent choice of grid size that makes
interpolation unnecessary. The step size of each dimension is
determined by the following procedure:
Initially viis assumed to be 0. Equations (III.1) and (III.2)
can then be reduced to
Δx=1
2aiΔt2(III.3)
Δv=aiΔt(III.4)
Choosing Δxand ΔtthegridsizeforΔvis found by
Δv=2Δx
Δt2Δt(III.5)
With this definition all resulting distances, xi, for a chosen
speed, vi, at time tifall on the defined grid.
Due to the fact that the three dimensions, (x,v,t),inthe
dynamic programming computation are dependent the initial
state might not be reachable by all paths within one final step
in the calculation. This becomes obvious looking at the last
step where xi,vi,xi+1,vi+1, and Δtare fixed by the computed
optimal paths up to the (i+1)th step and the initial conditions
(at time i). Given these values the resulting system of equations
consists of two equations with only one unknown, which is not
solvable. Hookers publication does not mention any adaptation
of his calculation to this fact. In our calculation it was found
that leaving the second to last iterative step (at t=1) free, and
independent of the grid, can ensure that all possible trajectories
computed can reach the initial state.
This process is demonstrated in the following. Expanding
Equations (III.1) and (III.2) for two time steps results in the
following four equations:
xi+1=xi+viΔt+1
2aiΔt2(III.6)
vi+1=vi+aiΔt(III.7)
xi+2=xi+1+vi+1Δt+1
2ai+1Δt2(III.8)
vi+2=vi+1+ai+1Δt(III.9)
Replacing xi+1and vi+1in Equations (III.5) and (III.6) with
Equations (III.3) and (III.4) results in
xi+2=(xi+viΔt+1
2aiΔt2)+(vi+aiΔt)Δt+1 2ai+1Δt2
(III.10)
vi+2=(vi+aiΔt)+ai+1Δt(III.11)
With these 2 Equations, which represent the second to last
and last time step in the dynamic programming process, the
two unknowns aiand ai+1can be computed with the fixed
variables xi,vi,xi+2,vi+2, and Δt. Using this concept all
possible trajectories can be explored for optimality.
0200 400 600 800 1000 1200
0
50
100
150
time
speed [km/h]
drive cycle
02000 4000 6000 8000 10000 12000
0
50
100
150
distance [m]
speed [km/h]
Maximum speed as function of distance
Figure 4. NEDC Drive Cycle with maximum speed limitations
B. Drive Cycle Generation
To quantify improvements due to eco-driving strategies a
baseline has to be defined for comparison. Using standard
and real-life drive cycles to represent average driving behavior
the differences in fuel consumption and vehicle operation
between general driving and eco-driving can be evaluated.
A similar approach to compare gains of optimal to general
vehicle operation was used by van Keulen in the development
of optimal energy management for a hybrid electric truck [15].
Using a drive cycle, where the velocity is defined as
a function of time, the constraints for an optimization are
defined. A maximum speed profile corresponding to the speeds
seen in the cycle is specified as a function of distance. Such a
maximum speed function can be defined by fixing speed limits.
Here the general limits of the roads in France were used and
consist of 30km/h, 50km/h, 70km/h, 90km/h, 110km/h and
130km/h zones. Ensuring that the vehicle stops at the same
distances as in the drive cycle an additional 0km/h limit is
introduced. Such a maximum speed function dependent on
distance can be seen in Figure 4. Here the velocity profile
of the NEDC (New European Drive Cycle) is shown in the
first plot. This speed profile is integrated over time to identify
the distance traveled. In the second plot in blue the speed
as a function of distance can be seen. In green the assigned
maximum speed limits are shown. It can be seen that at several
intermediate distances the vehicle is required to come to a full
stop. The idling times at each of these stops is added after the
optimization process.
In this analysis it is assumed that the vehicle has to cover
the same distance as specified by the drive cycle. Using the
cycle length a final time constraint can be defined as well. This
strategy should result in a fair comparison in fuel consumption
and vehicle operation between general driving, represented by
the drive cycle, and eco-driving, where the same distance with
the same stops is driven and the target is reached in the same
time.
IV. RESULTS
To evaluate the gains of an eco-driving strategy and to
understand the optimal operation of a conventional vehicle op-
timal operation of a Clio 1.5L vehicle was simulated using the
NEDC as base cycle. Using the previously defined distance,
time and maximum speed constraints the optimal speed profile
was calculated and can be seen in Figure 5 together with the
original cycle. Here the original NEDC cycle is displayed in
green, the maximum speed constraints can be seen in blue
and the computed optimal velocity for this trip is shown in

0 2000 4000 6000 8000 10000 12000
0
20
40
60
80
100
120
140
distance [m]
velocity [km/h]
cycle eco
max speed
NEDC
Figure 5. Preliminary Results: Drive cycle versus eco-drive cycle
0 200 400 600 800 1000 1200
0
20
40
60
80
100
120
time [s]
velocity [km/h]
Speed profil
0 200 400 600 800 1000 1200
0
1
2
3
4
5
time [s]
gear
Gear selection
eco cycle
NEDC cycle
Figure 6. Speed profile for NEDC cycle and eco-drive cycle
red. In this graph, the cycles are shown as a speed dependent
on distance whereas in Figure 6 speed is displayed with its
dependency on time. Due to the difference in vehicle speed
the eco-driver does not perform the same stops at the same
time but rather at the same distance.
Simulating the vehicle in a forward facing manner with the
help of the VEHLIB [11] software the operation of the vehicle
was compared for the original NEDC cycle and the eco-drive
cycle. The two speed trajectories with its assigned gears can
be seen in Figure 6, where the NEDC is plotted in green and
the eco-drive cycle in red. The fuel consumption of the Clio
over the NEDC is calculated to be 3.87L/100km while used
in the most efficient way the same vehicle for the same trip
in the same time can achieve a consumption of 3.26L/100km.
It can be seen from the graph that the vehicle uses short but
high acceleration phases to reach a rather low constant speed.
In the NEDC the vehicle acceleration lasts longer and reaches
a higher speed. Because of the fast acceleration of the eco-
driver the two vehicle can still arrive at the final distance at
the same time.
In the given NEDC drive cycle the gear choice is given
by the cycle. Utilizing this eco-driving strategy the gears are
chosen to be optimal. That means for a given wheel speed and
acceleration the gear that results in the least fuel consumption
is chosen. The difference between the gear choice in the initial
cycle and the eco-drive cycle is significant as demonstrated in
Figure 6 plot 2. In Figures 7 and 8 the operation of the engine
can be seen for the two cycles.
While the engine is operated mostly in the low torque
high speed region in the NEDC cycle in eco-driving the
engine operation is pushed to very high or maximum torque
operation. It should be noted here that maximum acceleration
in this simulation is only limited by the maximum engine
torque. The torque interrupts due to shifting are kept to a
0500 1000 1500 2000 2500 3000 3500 4000 4500
−50
0
50
100
150
Rotational Speed [rpm]
Torque [Nm]
Specific Engine Consumption [g/kWh]
210
210
215
215
215
225
225
225
250
250
250
275
275
275
300
300
300
400
400
400
Figure 7. Engine operation NEDC
0500 1000 1500 2000 2500 3000 3500 4000 4500
−50
0
50
100
150
Rotational Speed [rpm]
Torque [Nm]
Specific Engine Consumption [g/kWh]
210
210
215
215
215
225
225
225
250
250
250
275
275
275
300
300
300
400
400
400
Figure 8. Engine operation Eco-Drive Cycle
minimum such that the vehicle can follow the computed speed
trajectory. Although the engine operation changes, average
engine efficiency over the drive cycle was found to be almost
constant. The engine operates at an average 28.5% over the
NEDC cycle, while the eco-driver drives with an average
engine efficiency of 29.0%.
From these values it becomes obvious that the reduction in
fuel consumption of almost 16% does not only come from
an improvement in engine operation. This result shows that
the losses in the engine contribute to the overall losses of the
system but do not outweigh efficiency of the other components
in a vehicle. In the concept of eco-driving the fuel consumption
is reduced which is equivalent to maximizing overall efficiency
of the vehicle. Contributing to this efficiency is the engine but
also the drive train as well as the chassis of the vehicle.
In Table I the efficiency values for the drive train can be
found. As mentioned previously the engine efficiency has not
changed by a large amount. Final Drive efficiency remains
constant between the NEDC and the eco-drive cycle while
gearbox efficiency has improved by 0.3%. It can be concluded
from these values that the majority of losses reduced due to
eco-driving are not found in the drive train nor in the engine.
In Figure 9 the energy required by the vehicle to perform
each of the two cycles is displayed. Cycle 1 here represents
the NEDC cycle and cycle 2 stands for the derived eco-drive
cycle. In this plot the energy required to move the inertia of the
vehicle is shown in red, and the energy spent on overcoming
the resistance forces can be seen in green. It is illustrated in
NEDC eco-drive cycle
Engine efficiency [%] 28.5 29.0
Final Drive efficiency [%] 97.0 97.0
Gearbox efficiency [%] 97.9 97.6
Table I
EFFICIENCY OF DRIVE TRAIN FOR NEDC AND ECO-DRIVE CYCLE