Lane-free Traffic with Connected and Automated Vehicles - An
Optimal Control Approach
V. K. Yanumula∗ , P. Typaldos, D. Troullinos, M. Malekzadeh, I. Papamichail, and M.
Papageorgiou
Dynamic Systems and Simulation Laboratory, Technical University of Crete, Chania, Greece{karteek, ptypaldos, dtroullinos, mmalek, ipapa, markos}@dssl.tuc.gr∗ Corresponding author
Extended Abstract submitted for presentation at the 11th Triennial Symposium on
Transportation Analysis conference (TRISTAN XI)
June 19-25, 2022, Mauritius Island

March 26, 2022
Keywords: Lane-free traffic, connected and automated vehicles, optimal path planning, model
predictive control

1

INTRODUCTION

Inefficient usage of the road space is a major reason for traffic congestion, which entails increased
delays and fuel consumption and increased emissions. In addition, the vast majority of road accidents are caused due to erratic human driving. Despite extensive research on traffic management,
congestion and accidents occur excessively across the world. To mitigate these issues, connected
and automated vehicles (CAVs) seem promising (Montanaro et al., 2019). When the roads
are largely populated with CAVs, using Vehicle-to-Vehicle (V2V) and Vehicle-to-Infrastructure(V2I) information sharing, lanes may become unnecessary, as they waste lateral road space due
to variations in the width of vehicles, and necessitate accident-prone lane-changing manoeuvres.
With the lane-free traffic paradigm proposed by (Papageorgiou et al., 2021), the lane structureof traffic, designed to facilitate human driving, can be dropped in the era of CAVs in favourof increased safety and efficiency. In fact, the real-time perception capabilities of automatedvehicles are superb, compared to human drivers, due to a large number of sensors with 360◦
scanning and multiple scans per second. The CAVs, backed with appropriate path planningalgorithms, may move on a lane-free road, whereby vehicle movement may be influenced not
only by other vehicles in front of them (as with human driving), but also by vehicles behind,something that is referred to as ‘nudging’ (Karafyllis et al., 2020). The current work presents
an optimal control problem (OCP) approach for path planning, without the need for discrete
lateral positioning thanks to lane-free driving that renders lane changing obsolete. The dynamic
driving environment calls for frequent path planning updates over short time horizons (e.g. 8s), as long horizons inevitably entail outdated predictions on surrounding traffic. An efficientfeasible direction algorithm (FDA) (Papageorgiou et al., 2016) is used to solve (in 15.9 ms on
average) the OCP for each vehicle in a model predictive control (MPC) frame, which enables the
CAVs to use updated information.

2

OPTIMAL CONTROL PROBLEM (OCP)

Due to computer-based decision making in real time, the OCP consists of a discrete-time kinematic system model, with bounded control input, and an objective function, whose minimization
TRISTAN XI Symposium

Original abstract submittal

V. K. Yanumula, P. Typaldos, D. Troullinos, M. Malekzadeh, I. Papamichail, and M.
Papageorgiou

2

reflects a number of goals. More specifically, each ego vehicle (EV) is described by the following
set of equations, with real-valued positioning on both longitudinal and lateral directions, thanks
to the lane-free structure.
1
x1 (k + 1) = x1 (k) + T x3 (k) + T 2 u1 (k)21x2 (k + 1) = x2 (k) + T x4 (k) + T 2 u2 (k)2x3 (k + 1) = x3 (k) + T u1 (k)

(1a)
(1b)
(1c)

x4 (k + 1) = x4 (k) + T u2 (k)(1d)

where T is the time step; k is the integer time index, related to continuous time t via t = kT ;the states x1 , x2 are longitudinal and lateral positions of the vehicle centre, respectively; x3 , x4
are longitudinal and lateral speeds, respectively; while u1 and u2 are longitudinal and lateral
accelerations, respectively, i.e. the control inputs. The CAVs are essentially moving longitudinally, along a highway or arterial, with lateral road boundaries; consequently, the decoupledmodel (1) is sufficiently accurate, particularly under the influence of an appropriate term in theobjective function that discourages infeasible lateral steering. The upper and lower bounds on
the accelerations are as follows
umin1 (x3 ) ≤u1 (k) ≤ umax1(2a)

umin2 (x2 , x4 ) ≤u2 (k) ≤ umax2 (x2 , x4 )(2b)As seen, the longitudinal upper bound is constant, reflecting vehicle capabilities and passenger
convenience. The longitudinal lower bound and lateral bounds are state-dependent. Specifically,
the longitudinal lower bound is designed to prevent the vehicle from reaching negative speeds at
the next time step, i.e. to ensure x3 (k + 1) ≥ 0; from which, using state equation (1c), we obtainu1 (k) ≥ − T1 x3 (k). In addition, the lower bound needs to consider a comfortable limit UMIN1for smooth ride quality; hence, the longitudinal lower bound is defined as1
umin1 = max − x3 (k), UMIN1(3)
T
Vehicles are considered to be moving on unfolded roads that have straight lines as road boundaries. The CAVs should drive at times k and k + 1 so as to stay inside the road boundaries at
k + 2, i.e., ew ≤ x2 (k + 2) ≤ rw − ew , where rw is the road width and ew is the half width of EV.
If EV is moving on the road boundary, it should satisfy x4 (k + 2) = 0 to stay at least on the roadboundary. Thus, using again the state equations (1b) and (1d), the state-dependent bounds onlateral acceleration are as follows
−

1
3
1
3
[x2 (k) − ew ] −x4 (k) ≤ u2 (k) ≤ − 2 [x2 (k) − rw + ew ] −x4 (k)
2
T
2T
T
2T

(4)

The bounds in (4) represent a dead-beat controller with high magnitudes. Considering passenger comfort and vehicle capabilities, the controller gains are chosen more moderate, and thegeneralized representation is given as
umin2 (x2 (k), x4 (k)) = −Klat1 [x2 (k) − ew ] − Klat2 x4 (k)(5a)

umax2 (x2 (k), x4 (k)) = −Klat1 [x2 (k) − rw + ew ] − Klat2 x4 (k)(5b)

with 0 < Klat1 ≤ 1/T 2 and 0 < Klat2 ≤ 3/(2T ) as feedback controller gains chosen appropriately
to prevent road departures, as EV may reach the road boundaries asymptotically.

TRISTAN XI Symposium

Original abstract submittal

V. K. Yanumula, P. Typaldos, D. Troullinos, M. Malekzadeh, I. Papamichail, and M.
Papageorgiou

3

The objective function is made up of several sub-objectives to account for passenger comfort,
fuel consumption, target speed, collision avoidance and infeasible steering, as follows:
K−1
X

n
X

(

J=

2

2

2

2

w1 (u1 (k)) +w2 (u2 (k)) +w3 [x3 (k)−vd1 ] +w4 [x4 (k)−vd2 ] +w5)[ci (x, oi )]+w6fc

(6)

i=1

k=0

The first two terms aim for passenger comfort and also for minimization of fuel consumption.
The third and fourth terms help the EV to reach its target speeds, vd1 and vd2 in longitudinaland lateral directions, respectively. An interaction zone with equal length on upstream anddownstream directions is defined to communicate with other vehicles that are treated as obstacles(OVs). The length of the interaction zone equals to the product of the longitudinal desired speed
and the planning horizon K. For n OVs with relative position of their centers (oi1 , oi2 ) inside
theP
interaction zone and with speeds (oi3 , oi4 ), the obstacle avoidance sub-objective is defined as
w5 ni=1 [ci (x, oi )], where, for ith OV, we use an ellipsoid-like function,
ci (x, oi ) = 1 − tanhx1 − δ o1
r1

p1


+

x2 − oi2
r2

 p2 + h

x1 −δo1
r3

p 3

+

1


x2 −oi2
r4

p4 ip5(7)+1

with exponents p1 to p5 influencing the ellipsoid shape. The ellipsoid function (7) is rather
complex, as it needs to adequately address two requirements. Firstly, its iso-cost curves aroundthe vehicle should approximate and fully cover the rectangular vehicle shape, without wasting
too much space around the rectangle; this is achieved by the first term in (7) with appropriate
values for the exponents p1 and p2 . However, that term by itself exhibits a very flat shape at
the interior of the rectangle, which implies accordingly low gradient values and therefore slow
convergence of the numerical solution algorithm for the optimal control problem. Therefore, thesecond term is added, which does not alter the iso-cost curves around the vehicle, but provides
a steep increase of the overall function (hence strong gradient values) at the interior of thevehicle-rectangle. Moreover,r1 = 0.5 (Li + ωx1 x3 + ωx1 oi3 )

q
h
i
2
r2 = 0.5 Wi + ωx2 tanh(oi2 − x2 ) (x4 − oi4 ) + [tanh(oi2 − x2 ) (x4 − oi4 )] + w

(8a)
(8b)

where, Li = le + loi with le and loi being 1.3 times the lengths of EV and OV; Wi = we + woi
with we and woi being 1.2 times the widths of EV and OV; ωx1 and ωx2 are time-gap likeparameters. The ellipsoid is positioned longitudinally at δo1 = oi1 − ωx1 (x3 − oi3 ) /2; r3 = 0.5r1
and r4 = 0.5r2 . To visualize, consider EV positioned at (10, 4) m and an OV at (30, 5.1) m,with length of each equal to 5 m, width equal to 1.8 m; longitudinal speed of EV at 35 m/s andof OV at 30 m/s. The contour plot of (7) with p1 = 6, p2 = p3 = p4 = p5 = 2, ωx1 = 0.35,
ωx2 = 0.5 is shown in Figure 1, in which, the red box represents the physical dimensions (Li ,Wi ), while, the green box represents ellipsoid dimensions. It may be seen that the ellipsoid may
cause both repulsing and nudging influence to other vehicles.
0.039

0.312

0.585

0.858

1.131

1.404

1.677

40

50

1.950

10
8
6
4
2
0
0

10

20

30

60

Figure 1 – Sample of OV ellipsoid.
TRISTAN XI Symposium

Original abstract submittal

V. K. Yanumula, P. Typaldos, D. Troullinos, M. Malekzadeh, I. Papamichail, and M.
Papageorgiou

4

In order to discourage the vehicle from taking infeasible values of steering, the following subobjective that increases quadratically if the lateral speed exceeds β times the longitudinal speed,for some small β > 0 is considered,2
(βx3 (k) − x4 (k)) if x4 (k) > βx3 (k)fc = (βx3 (k) + x4 (k))2 if x4 (k) < −βx3 (k)
(9)


0
otherwise
The sub-objective function for obstacle avoidance may not fully guarantee collision-free movement, which calls for additional measures in the rare case that the generated trajectory includes
a collision. Thus, each OCP trajectory generation is checked for possible collisions. If that is
detected, the OCP is resolved with stricter bounds on the control input. In case of longitudinal
collision, the ego vehicle uses the rear bumper position of that OV (plus a time gap) as a moving
boundary. Then, the state-dependent lower and upper bounds of the longitudinal acceleration
are re-defined, similarly to lateral bounds, as,1
umin1 = max − x3 (k), UMIN11
(10a)
T

(10b)
umax1 = max umin1 , Klong1 (−x1 (k) + x1Lim ) + Klong2 (oi3 (k) − x3 (k)) + u1o (k)where, x1Lim = oi1 (k) − l2e − loi
2 − ωx1 emer x3 (0) and UMIN11 is an emergency value to allow
the EV to decelerate harder. On the other hand, in case of a lateral collision, the movementof EV is restricted laterally by assuming virtual road boundaries that form a narrow corridor
around the current lateral position of the EV, while the bounds on lateral acceleration remainunchanged.

3

RESULTS AND CONCLUSION

Extended simulations were performed using the TrafficFluid-Sim (Troullinos et al., 2021), an
extension designed for lane-free traffic to be used with the SUMO (Simulation of Urban MObility)
simulator (Lopez et al., 2018). All the vehicles are assumed to be moving on an unfolded ringroad of 1.0 km length and 10.2 m width, with eight different vehicle sizes, and target speeds
randomly assigned between 25 and 35 m/s. The flows achieved during the last ten minutes of
1-hour simulations for different densities are given in Table 1. On a road width that is equivalent
to three conventional lanes, high flow values are reached on the account of lane-free environment.
Table 1 – Flow outcomes for given densities.
Density (veh/km)
Flow (veh/hr)

50
100
150
200
250
300
350
5354 10464 14899 18381 15813 13003 10615

400
8156

450 500
5707 2878

Videos reflecting vehicle driving on the ring-road may be viewed at https://bit.ly/TF-OPP. The
optimal path planning for CAVs seems promising and delivers good results. Multiple OCPs are
solved at each time in an MPC framework with event-based updating to generate finite horizon
trajectories for respective EVs. The proposed method results in high flows, small computation
times and smooth trajectories for passenger comfort. More results are omitted due to lack ofspace and will be included in the presentation and full paper. Related work is going on toconsider ramps, variable road widths and complex networks.
Acknowledgement
This research received funding from the European Research Council under the EU Horizon 2020
Programme / ERC Grant no. 833915, project TrafficFluid, see: https://www.trafficfluid.tuc.gr
TRISTAN XI Symposium

Original abstract submittal

V. K. Yanumula, P. Typaldos, D. Troullinos, M. Malekzadeh, I. Papamichail, and M.
Papageorgiou

5References
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Original abstract submittal