Deriving Spatial Policies for Overtaking Maneuvers with Autonomous Vehicles

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Deriving Spatial Policies for Overtaking Maneuvers
with Autonomous Vehicles
Jayanth Bhargav
Electrical & Systems Engineering
University of Pennsylvania
Philadelphia, USA
jayanthb@seas.upenn.edu
Johannes Betz
Electrical & Systems Engineering
University of Pennsylvania
Philadelphia, USA
joebetz@seas.upenn.edu
Hongrui Zehng
Electrical & Systems Engineering
University of Pennsylvania
Philadelphia, USA
hongruiz@seas.upenn.edu
Rahul Mangharam
Electrical & Systems Engineering
University of Pennsylvania
Philadelphia, USA
rahulm@seas.upenn.edu
Abstract—Planning an accurate and safe trajectory is a crucial
element in autonomous driving. To execute complex driving ma-
neuvers like overtaking, motion planning requires an enhanced
decision-making algorithm that decides the when, where and how
of the overtaking maneuver. This paper proposes an algorithm
that increases the likelihood of a safe overtaking maneuver by
learning spatial information. Here, spatial information refers to
the track portion/curve and the position of the ego vehicle with
reference to that. The technique is applied to an autonomous
racing setup where vehicles have to detect and operate at the
limits of dynamic handling. To learn the spatial information,
offline experiments of a 2-player race are conducted to generate
probability distributions of overtaking maneuvers conditioned on
speed and relative-position of the ego vehicle with respect to the
opponent. Furthermore, a Switched Model Predictive Contouring
Controller (SMPCC) is proposed for incorporating the policy
learning algorithm into the path planning and control setup.
Extensive simulations show that the proposed algorithm is able
to achieve an increased number of overtakes at different track
portions on known and unknown race tracks.
Index Terms—autonomous systems, automobiles, intelligent
vehicles, optimal control, path planning
I. INTRODUCTION
A. Autonomous Racing
Autonomous racing has become popular over the recent
years and competitions like Roborace [1] or the Indy Au-
tonomous Challenge as well as small-scale competitions like
F1Tenth [2] provide platforms for evaluating autonomous
driving algorithms and software. The overall goal of all these
competitions is that researchers and engineers can develop
algorithms that operate vehicles at the edge: high speeds,
high accelerations, high computation power, adversarial en-
vironments. The algorithms that were developed in the field
of autonomous racing so far are mostly focusing on single
vehicle only that try to achieve a human-like lap time. The
field of high dynamic overtaking maneuver with dynamic
opponents are less displayed. In addition, achieving a human-
like behavior (e.g. like a Formula 1 race driver) that makes
the decision about an overtaking maneuver and executes a
secure and reliable maneuver at high speeds is still an unsolved
problem.
B. Contributions
In this paper, an approach to learn spatial information for
overtaking maneuvers in autonomous vehicles is presented.
This work has three primary contributions:
1) Design of Experiments (DoE) for offline policy learning.
2) An application of autonomous driving to learn effec-
tive overtaking maneuvers for autonomous race cars.
Discretization of selected f1 tracks into a category of
turns/curves and simulations of 2-player race to derive
overtaking policies for different track portions.
3) A Switched Model Predictive Contouring Controller
(SMPCC) setup based on [3], which combines a re-
ceding horizon control algorithm and specific driving
behaviours.
II. RELATED WORK
Dixit et al. [4] provide a state of the art review of trajectory
planning and control for autonomous overtaking maneuvers.
The authors state finally in their review, that two important
aspects of trajectory planning for high-speed overtaking need
to be addressed: (i) inclusion of vehicle dynamics and en-
vironmental constraints and (ii) accurate knowledge of the
environment and surrounding obstacles.
Although the state of the art displays a plethora of al-
gorithms for path and behavioral planning of autonomous
vehicles, explicit algorithm development for autonomous race
cars is relatively lesser. As part of the Roborace competition
[5], [6] [7] presented a planning and control system for real life
autonomous racing cars. Both approaches focused on a holistic
software architecture that is capable of dynamic overtaking.
Nevertheless none of them realized a head to head race with
the vehicles. As a part of the same competition, [8] presented
a nonlinear model predictive control (NMPC) for racing. The
overtaking strategy was implemented as a term in the objective
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function. The NMPC has the freedom to choose the side
for an overtake and was mainly relying on the obstacles
velocity to perform the overtaking maneuver. In [9] a simple
Q-Learning algorithm is applied to learn the behavior of an
virtual opponent car to apply an effective overtaking strategy
on either the straight or right before tight bend.
In [10] a method to plan overtaking maneuvers in autonomous
racing based on gaussian processes is presented. This machine
learning method is able to learn the behavior of the opponent
vehicle. Based on the outputs of this process, a stochastic MPC
plans optimistic trajectories that lead to a controlled overtaking
maneuver of the lead vehicle.
In multi vehicle racing, [11] presented a non-cooperative game
theory approach where autonomous racing, formulated as rac-
ing decisions is a non-cooperative nonzero-sum game. Liniger
et al. [11] displayed that different games can be modelled that
achieve successfully different racing behaviors and generate
interesting racing situations e.g. blocking and overtaking.
Notomista et al. [12] considered a two-player racing game
where the ego vehicle is based on a Sensitivity-ENhanced
NAsh equilibrium seeking (SENNA) method, which uses an
iterated best response algorithm in order to optimize for a
trajectory in a two-car racing game. Jung et al. [13] present a
game-theoretic MPC approach for head-to-head autonomous
racing that consists of a (1) game-based opponents’ trajec-
tory predictor, (2) high-level race strategy planner, and (3)
MPC-based low-level controller. Based on the results of the
prediction, the high-level race strategy planner plans several
behaviors to respond to various race circumstances.
The state of the art displays that the autonomous rac-
ing community is focusing on integrating effective learning
techniques and strategies into dynamic path and behavioral
planning to make the car faster, more reliable and more in-
teractive [14] [15] [16]. Improvements in planning/control for
overtaking maneuvers have not yet been explored extensively
and learning from spatial information (track portions and
position of the vehicle on the track) has not been examined
before.
III. DESIGN OF EXPERIMENTS
We propose an offline experiment setup which will create
emphasis on specific track portions and examine the overtaking
maneuvers. With these offline experiments, it is possible to
create track-based policies that can be used in a high level
decision maker, behavior or motion planner.
A. Track Portions
Based on our racetrack application, in the first step we
define the track portions that we will examine. We will use
four high level definitions of track portions that are the most
common kinds of curves/turns found on racetracks: Straight,
Sweeper Curve,Hairpin Curve and Chicane
For example, consider the racetrack in Budapast, Hungary.
In figure 1 we display 11 different track portions on the track
that are marked with labels 1 to 11.
Fig. 1. Example racetrack for offline experiments: Budapest Circuit, Hungary
Track portions are defined by T={τ|1≤τ≤11, τ ∈N}
and are categorized in the four track segment types:
•Straight: 11
•Sweeper Curve: 3, 7, 8
•Hairpin Curve: 1, 2, 4, 9, 10
•Chicane: 5, 6
B. Sampling based trajectory rollouts
To examine these defined track portions we setup an offline
simulation that varies different parameters visualized in figure
2.
Lateral Variation
Longitudinal
Variation Opponent car
On optimal raceline
Overtaking
Detection Zone
Fig. 2. Ego vehicle (red car) starting behind the opponent vehicle (blue car)
on the track. Apart from the velocity, lateral and longitudinal positions of the
ego vehicle are varied as shown in the figure. The opponent vehicle follows
a pre-computed race line and is non-interactive. The overtaking maneuver is
examined in the overtaking detection zone (marked in yellow).
The opponent vehicle follows a curvature optimal pre-
computed race line based on [17] and is non-interactive.
For every track portion τ∈T, a uniformly sampled set
of positions P:XXY ⊂ R2are chosen as the starting
position for the ego. The obstacle vehicle speed is varied as
vobs =vbaseline ∗(1+ s)where s∈ {−0.2,0,+0.2},vbaseline
being the speed of the obstacle from the pre-computed optimal
race line.
The agents are initialised with these positions and set to
start the simulation. The ego vehicle synthesises dynamic tra-
jectories based on the MPCC planner with obstacle avoidance.
A fully observable model is used for the ego vehicle i.e. the
ego vehicle will have the information of the track portion τ
which it is driving in and the current state of the obstacle
Xobs = [xobs, yobs , ϕobs]
In this setup, we conduct simulations based on the following
parameter variations:
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•Lateral offset: The position of the ego vehicle is varied
lateral across the track with an offset from the centerline.
•Longitudinal offset: The position of the ego vehicle is
varied longitudinal along the centerline of the track.
•Opponent speed change: The opponent speed is varied
with ±20% from baseline
With a high expectation, the ego vehicle will succeed in an
overtaking maneuver when the obstacle speed is 20% lower
than its baseline. This verifies the fact that speed advantage
always helps in overtaking (e.g. DRS zones in F1). The next
set of parameters that influence the overtaking maneuver is
the position. In convoluted race tracks, we can display that
starting off at a specific position gives us a higher chance of
an overtaking maneuver. For each track portion, we define four
regions of interest: R1,R2,R3and R4. Starting positions of
the ego vehicle are uniformly sampled in all the four regions
to generate experimental data.
IV. PLANNING AND CONTROL SETUP
Continuous time system dynamics is used to develop a
constrained optimal controller to steer the vehicle in the
track. The optimal planner plans the path for a horizon of N
steps ahead, steers the vehicle with the first step, and again
repeats the process for the specified amount of time. This is
a modified form of the Model Predictive Controller (MPC).
A. Model Predictive Contouring Control
The MPCC problem defined in [3] is re-formulated into a
finite-continuous time optimal control problem as follows:
min ZT
0ϵlin
c(t)ϵlin
l(t)Qc 0
0Ql ϵlin
c(t)
ϵlin
l(t)
−Qθ˙
θ(t) + uT(t)Ru(t)dt
s.t. ˙x=f(x, u, Φ)
blower ⪯x(t)⪯bupper
llower ⪯u(t)⪯lupper
h(x, Φ) ≤0
given the system dynamics fand the arclength parametriza-
tion of the contour (the track) Φ. A single-track bicycle model
is used. Here x(t)denotes the system state, u(t)the inputs
to the system, bthe box constraints on the state, lthe box
constraints on the input and hcaptures the track boundary
constraints. The state of the system is augmented with the
advancing parameter θand the virtual input ˙
θis appended to
the inputs from the original system dynamics.
Qc,Ql,Qθand Rare the cost-function parameters of the
MPC controller.
The track boundary constraint is realized as a convex disk
constraint.
h(x, Φ) = x−xlin
t(θ)2+y−ylin
t(θ)2−rΦ(ˆ
θ)2
Here rΦ(ˆ
θ)is the half-width of the track at the last predicted
arc length.
The contouring error ϵlin
cand lag error ϵlin
ldescribed in [3]
are modified by linearizing them around the previous solution
θ.
The MPCC is optimizing to move the position of a virtual
point θ(t)along the track to achieve as much progress as
possible while steering the model of the vehicle to keep
contouring and lag errors as small as possible.
The center-line of the track is given in way-points (X-and Y-
position). To implement MPCC an arc-length parametrization
Φis required. This is realized by interpolating the way-points
using cubic splines with a cyclic boundary condition, and
creating a dense lookup table with the track location and the
linearization parameters.
B. Switched Model Predictive Contouring Control (SMPCC)
To achieve more control over the path planning of the ego
vehicle, the proposed SMPCC setup is displayed in figure 3.
Agent with SMPCC Planner
Normal Mode Drive Right Drive Left
−t<ec< t −ϵ<ec< t t<ec< ϵ
Fig. 3. Modes of SMPCC
In this, the agent switches between different modes defined
by different solver formulations. The constraints for the modes
as shown in fig. 3 are added to the problem formulated
in section IV-A. These constraints are tuned with the slack
variable (ϵ) to ensure that the planner does not get stuck into
an in-feasibility loop leading to a crash. The ego can therefore
overtake on both left and right side of the opponent vehicle.
The MPCC control problem is solved by efficient interior point
solvers in FORCES [18].
V. RESULTS AND DISCUSSION
A. Offline Spatial Policy Learning
In the following section, we present the results from the
offline experiments. Algorithm 1 elucidates the offline exper-
iment based policy learning developed in this paper.
X,Yare the set of x and y coordinate offsets (expressed
as percentage of track width) and S={−0.2,0,+0.2}is
the speed offset expressed as a percentage change from the
baseline obstacle speed.
The obstacle update model is g(·), which is a pre-computed
curvature optimal race line of the race track under consider-
ation. Define Ψ:{Silverstone circuit (England), Hungaroring
circuit (Budapest), Catalunya circuit (Spain) and N¨
urburgring
circuit (Germany)}, the set of race circuits on which the race
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is conducted for learning the policies. In total we simulate
576 experiments based on the 16 lateral, 12 longitudinal and
3 velocity variations for each track portion in each of the
four racetracks. The algorithm 1 populates the policy map Πk
with the track regions for each of the curves, having highest
probability of overtakes for all k∈Ψ.
Algorithm 1: Offline Spatial Policy Learning
Function MPCC Planner(Xobs):
solve MPCC problem defined in Section IV
return u∗;
Function Check Overtake(Xobs,Xego ):
project Xobs,Xego as s1, s2on track
return bool(s1> s2);
for k∈Ψdo
initialize: Πk={}
for τ∈Tdo
initialize: p = {}, overtakes = {}, total = {}
for x, y, s ∈ X XYXSdo
initialize: Xego, Xobs
for t= 0 to Tsim do
u∗=MPCC Planner (Xobs)
steer ego: X+
ego =f(Xego , u∗)
update obstacle pos: X+
obs =g(Xobs)
Xobs, Xego =X+
obs, X +
ego
identify track region i∈ {0,1,2,3}
if Check Overtake(Xobs, Xego )then
overtakes[Ri]++;
total[Ri]++ ;
end
end
compute p[Ri] = overtakes[Ri]/total[Ri],∀i
Πk[τ] = argmax(p)
end
end
Figure 4 describes an example of the four track regions
and the overtaking probabilities that are evaluated based on
algorithm 1. The overtaking probabilities for two of the
racetracks with their respective track portions are displayed
in tables I and II.
Track Region R1
Front Left
p(R1)= 0.43
Rear Left
Track Region R4
p(R4) = 0.23
Overtaking Area R2
Front Right
p(R2) =0.38
Track Region R3
Rear Right
p(R3) =0.28
Fig. 4. Predefined track regions of interest for overtaking maneuver at a
specific turn with overtaking success probabilities.
Results from the experiments conducted on the four race-
tracks are summarized in figures 5, 6 and 7. They display the
overtaking probability distribution for each overtaking zone
TABLE I
OVERTAKING PROBABI LITIE S FOR ALL T RACK PORTIONS - RACET RAC K 1
( SILVERS TONE, ENGLAND)
Track
Portion
(τ)
Track
Portion
Type
p(R1)p(R2)p(R3)p(R4)
1 Sweeper 0.99 0.93 0.52 0.59
2 Hairpin 0.63 0.52 0.31 0.33
3 Hairpin 0.41 0.38 0.25 0.21
4 Sweeper 0.65 0.67 0.57 0.59
5 Chicane 0.25 0.21 0.14 0.21
6 Straight 0.99 1.0 0.95 0.99
7 Sweeper 0.47 0.52 0.33 0.31
8 Hairpin 0.40 0.36 0.37 0.38
TABLE II
OVERTAKING PROBABI LITIE S FOR ALL T RACK PORTIONS - RACET RAC K 2
(BUDAPEST, HUN GARY)
Track
Portion
(τ)
Track
Portion
Type
p(R1)p(R2)p(R3)p(R4)
1 Hairpin 0.52 0.49 0.34 0.29
2 Hairpin 0.31 0.43 0.27 0.21
3 Sweeper 0.71 0.62 0.54 0.59
4 Hairpin 0.67 0.53 0.39 0.41
5 Chicane 0.41 0.43 0.22 0.25
6 Chicane 0.27 0.21 0.19 0.14
7 Sweeper 0.67 0.64 0.35 0.37
8 Sweeper 0.59 0.48 0.40 0.36
9 Hairpin 0.44 0.58 0.35 0.29
10 Hairpin 0.65 0.60 0.44 0.38
11 Straight 0.97 0.99 0.95 0.94
(R1-R4) at chicanes, sweeper curves and hairpins respectively
across all the race-tracks considered for policy learning. From
statistical analysis on the offline experiments for four different
racetracks and their 39 track portions, we have the following
observations:
•An overtaking maneuver will be more successful if we
are closer to the opponent vehicle. This can be seen in
both the raw numbers from table I and II as well in the
higher median and maximum in the boxplots for R1&
R2.
•On the straight, it does not really matter which side we
are on the track when trying to overtake, we just need
stay closer to the opponent.
•The sweeper curve generally has a high overtaking prob-
ability due to the high speeds of the car. We only get
an advantage here if we are close enough to the car and
therefore we need to be in region R1or R2.
•We can see that in each hairpin we have the highest
overtaking probability in either in R1or R2depending
on the nature of the hairpin turn: right or left. This is
due to the fact that being on the inside of the curve near
a hairpin, the car is able to achieve a better trajectory
through the hairpin.
•Achicane generally has the lowest overtaking probability
due to the fact that it is a complex region for the car to
maneuver and with lesser space for overtaking. Since the
chicane is a highly convoluted turn, curvature direction
does not indicate any better start regions.
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Front Left Front Right Back Right Back Left
0.4
0.6
0.8
1.0
Overtaking Probability
Fig. 5. Overtaking probability distribution for sweeper curves.
Front Left Front Right Back Right Back Left
0.2
0.3
0.4
0.5
0.6
Overtaking Probability
Fig. 6. Overtaking probability distribution for hairpins.
B. Online Policy Execution
Algorithm 1 furnishes a policy map for all the different
curves/track portions from different race tracks considered
during the policy learning phase. We will now compute the
best policy Π(s)as a function of the track portion sand
integrate it into the SMPCC setup on the ego vehicle. We then
compare the number of overtakes with and without the spatial
policy based MPCC controller to verify the effectiveness of
our algorithm.
Algorithm 2: Online Evaluation with SMPCC
Function SMPCC(Xobs, mode):
if mode = ‘normal‘ then
u∗=solve MPCC problem in Section IV
elsemodify MPCC to integrate policy (see fig. 3)
u∗=solve modified MPCC problem
end
return u∗;
initialize Xego,Xobs mode = ‘normal‘
for t= 0 to Tsim do
u∗=SMPCC (Xobs,mode)
steer the ego: X+
ego =f(Xego , u∗)
update obstacle position: X+
obs =g(Xobs)
Xobs, Xego =X+
obs, X +
ego
identify track portion τwhere ego is present
policy lookup: mode = Π[τ]
end
The results display that the offline policy learning approach
has been successful by showing an increased number of
Front Left Front Right Back Right Back Left
0.2
0.3
0.4
0.5
0.6
Overtaking Probability
Fig. 7. Overtaking probability distribution for chicanes.
TABLE III
RESULTS: NUMBER OF OVERTAKES WITH AND WITHOUT POLICY ON
RACETRACK 1 (SILVE RSTONE, EN GLAND )
Track
Portion (τ)Track
Portion
Type
Number of
Overtakes
Policy OFF
Number of
Overtakes
Policy ON
1 Sweeper 436 452
2 Hairpin 256 337
3 Hairpin 308 426
4 Sweeper 342 357
5 Chicane 117 302
6 Straight 565 566
7 Sweeper 237 283
8 Hairpin 218 394
TABLE IV
RESULTS: NUMBER OF OVERTAKES WITH AND WITHOUT POLICY ON
RACETRACK 2 (BUDAPEST, HUNGA RY)
Track
Portion (τ)Track
Portion
Type
Number of
Overtakes
Policy OFF
Number of
Overtakes
Policy ON
1 Hairpin 286 375
2 Hairpin 297 404
3 Sweeper 410 446
4 Hairpin 337 398
5 Chicane 270 361
6 Chicane 180 329
7 Sweeper 372 438
8 Sweeper 239 297
9 Hairpin 288 326
10 Hairpin 301 374
11 Straight 558 563
overtaking maneuvers at all racetracks and track portions.
We can observe that overtaking on the straights is usually
easy (even without policy). Since sweeper curves are usually
wide track portions that allow high speeds and do not involve
complicated maneuvers, both with and without switching
policy, we achieve higher overtaking maneuvers. Although
we see that the switching policy leads to more overtaking
maneuvers because having the right position for the overtaking
maneuver is crucial here, too. We see the highest impact of our
algorithm at hairpins and chicanes. This is mainly due to the
fact that overtaking at these curves is usually complicated and
needs a good strategy beforehand. We see that our algorithm
can nearly double the amount of overtaking maneuvers in the
chicane (track portion 10 of Catalunya) which substantiates the
fact that having the right starting position for an overtaking
maneuver is indispensable.
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C. Evaluation on an Unknown Track
As an ultimate test, we now apply this online policy
algorithm to the agent racing in an unknown racetrack (Sakhir
Circuit, Bahrain). The results from Sakhir Circuit show an
increase in the number of overtakes with the policy at all track
portions as displayed in table V.
TABLE V
RESULTS: NUM BER OF OVE RTAKE S WITH AN D WITHO UT POLICY ON
UNKNOWN RAC ETRACK 5 (SAKHIR, BAHRAIN)
Track
Portion
(τ)
Track
Portion
Type
Turn Di-
rection
(Left/
Right)
Policy
Region Number
of Over-
takes
Policy
OFF
Number
of Over-
takes
Policy
ON
1 Chicane Right R2254 298
2 Sweeper Right R1278 367
3 Chicane Left R1214 391
4 Hairpin Right R1319 421
5 Hairpin Left R2224 327
6 Straight Left R2551 560
7 Sweeper Right R1348 418
8 Sweeper Right R1297 368
9 Straight Left R2559 549
10 Sweeper Right R1311 396
11 Straight Right R1552 561
Additional offline policy evaluations have shown that a
generalization from turn directions and overtaking zones is
only partially useful. The generalised policies did not lead to
successful overtakes always and were not feasible in some
cases. A possible refinement could be the focus on using a
parametric curvature of the turn rather than the high level
definition of left or right.
VI. CONCLUSION AND FUTURE WORK
In this paper, an algorithm for spatial policy learning from
offline experiments is proposed to learn effective overtaking
strategies based on position advantage. Extensive simulations
on real world racetrack layouts show that the proposed al-
gorithm is able to learn regions of high probabilities on a
racetrack for successful and safe overtaking maneuvers. The
(SMPCC) setup, that has the driving policies integrated into
the motion planning and control stack of the vehicle resulted in
an increase in the number of overtakes. Specifically, the policy
based algorithm was found to be highly effective for convo-
luted track portions like chicanes, where a positional advantage
plays a major role in a successful overtaking maneuver. In
summary, with the setup defined in this paper, one can create
more realistic and better overtaking maneuvers for autonomous
vehicles. This brute-force technique of learning spatial infor-
mation serves as a fundamental result and ground truth for
future work. Extensions to this work will include learning-
based algorithms based on reinforcement learning techniques
to identify the overtaking probability based on the curvature
information of upcoming turns and can therefore applied to
behavioral planners for passenger autonomous vehicles. One
can consider non-reactive, reactive and aggressive opponents
which are defensive and sophisticated to overtake and therefore
deriving a holistic strategy for overtaking on e.g. highways.
With this setup, one can learn and integrate complex human-
like overtaking maneuvers for autonomous vehicles in a safe
and reliable manner.
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