The imminent pervasive deployment of connected and autonomous vehicle (CAV) technology offers various opportunities for new applications. Particularly, the cooperative behavior capability of CAVs is invaluable for critical operations within the transportation system.In this paper, we present an approach for cooperativeCAVs to support the navigation of emergency vehicles in the traffic. We first lay out the trajectory planning problem for an emergency vehicle and formalize it in terms of the locations and accelerations of vehicles in a certain range of the emergency vehicle. Then, this problem is encoded onto a graph, where we map the safety constraints using the edge weights. We show that cooperative planning of the surrounding vehicles can be accomplished by an optimization problem, where the objective is to maximize the second largest eigen-value of the resulting graph Laplacian. The simulation study demonstrates the effectiveness of the proposed system in terms of the resulting trajectories and the corresponding perturbation in the whole system.
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Navigation of Emergency Vehicles UsingCooperative Autonomous Driving
1. Problem context Mapping onto graphs Graph spectral properties Problem formalization Results
Cooperative autonomous navigation of
emergency vehicles
Harish Chintakunta1 Mustafa หIlhan Akbaยธs2
1Department of Electrical and Computer Engineering
Florida Polytechnic University
2Electrical, Computer, Software and Systems Engineering
Embry-Riddle Aeronautical University
Harish Chintakunta1
, Mustafa หIlhan Akbaยธs2
FPU & ERAU
Cooperative autonomous navigation of emergency vehicles
2. Problem context Mapping onto graphs Graph spectral properties Problem formalization Results
Emergency vehicle navigation
Making way for the EV can take a
long time.
Drivers feel โhassledโ at the
appearance of EVs, which leads
to bad decisions.
There are about 6,500 annual
accidents involving ambulances.
Harish Chintakunta1
, Mustafa หIlhan Akbaยธs2
FPU & ERAU
Cooperative autonomous navigation of emergency vehicles
3. Problem context Mapping onto graphs Graph spectral properties Problem formalization Results
Advantages of Connected Autonomous Vehicles (CAVs)
Vehicles can be noti๏ฌed well in advance about an
approaching EV.
Complex cooperative behavior amongst CAVs can assist
the navigation of EVs.
EV itself being autonomous can enable complex path
planning algorithm to optimize travel time, saving lives.
Harish Chintakunta1
, Mustafa หIlhan Akbaยธs2
FPU & ERAU
Cooperative autonomous navigation of emergency vehicles
4. Problem context Mapping onto graphs Graph spectral properties Problem formalization Results
Our view of the problem
We now consider the problem
when all other vehicles drive
cooperatively to assist an EV.
Facilitating safety for EV is
rephrased using topological
features.
The space of feasible paths
for EV should be โstrongly
connectedโ, in other words,
no โbottle necksโ.
Harish Chintakunta1
, Mustafa หIlhan Akbaยธs2
FPU & ERAU
Cooperative autonomous navigation of emergency vehicles
5. Problem context Mapping onto graphs Graph spectral properties Problem formalization Results
Solution strategy
1 Capture the notion of "topological connectedness" in terms
of an eigenvalue of a matrix.
2 Move the surrounding vehicles in order to increase this
eigenvalue.
3 The above process increases the topological
connectedness thereby increasing the safety.
Harish Chintakunta1
, Mustafa หIlhan Akbaยธs2
FPU & ERAU
Cooperative autonomous navigation of emergency vehicles
6. Problem context Mapping onto graphs Graph spectral properties Problem formalization Results
Formulation in terms of graph theory
1 Discretize the space and draw
edges between neighboring vertices.
2 Encode the safety information onto
edge weights.
3 The weight we = 1 โ eโd(s,e)
on an
edge is a decreasing function of the
distance from a surrounding vehicle.
4 In this example, we would want the
surrounding vehicle to move away
from the central edge.
Harish Chintakunta1
, Mustafa หIlhan Akbaยธs2
FPU & ERAU
Cooperative autonomous navigation of emergency vehicles
7. Problem context Mapping onto graphs Graph spectral properties Problem formalization Results
Inspiration from spectral properties
Graph Laplacian L de๏ฌned as
L = D โ A,
has the following useful property (M.Fiedler, 1973):
ฮป2(L) โค ฯe(G),
where ฯe(G) is the edge connectivity.
More importantly, in weighted graphs, ฮป2(L) is sensitive to
weights of the edges in an edge cut set.
Harish Chintakunta1
, Mustafa หIlhan Akbaยธs2
FPU & ERAU
Cooperative autonomous navigation of emergency vehicles
8. Problem context Mapping onto graphs Graph spectral properties Problem formalization Results
Properties of ฮป2(L)
1 ฮป2(Lฮ) is very sensitive to
weights of edges which
โconnectโ distinct
components.
2 The gradient of ฮป2(Lฮ) w.r.t
the elements of L can be
analytically expressed.
3 Numerical computation of the
gradient of ฮป2(Lฮ) also tends
to be accurate even for large
Lฮ matrices, making it ideal
for numerical optimization
algorithms.
โ0.2 โ0.1 0.0 0.1 0.2
โ1.00
โ0.75
โ0.50
โ0.25
0.00
0.25
0.50
0.75
1.00
0 1
2
3
4
5
6
7
8
9
10
11
12 13
14
15
16
17
18 19
0.2 0.4 0.6 0.8 1.0
weight of the linking edge
0.0025
0.0050
0.0075
0.0100
0.0125
0.0150
0.0175
secondeigenvalue
linking edge
insider edge
Harish Chintakunta1
, Mustafa หIlhan Akbaยธs2
FPU & ERAU
Cooperative autonomous navigation of emergency vehicles
9. Problem context Mapping onto graphs Graph spectral properties Problem formalization Results
ฮป2(L) as a function of vehicle position
1 ฮป2(L) gets smaller when
the vehicle approaches the
critical edge.
2 If fact, the gradient
x ฮป2(L) consistently
points away from the
central edge.
3 In this choice of the weight
function, the value is
independent of the position
x, given x is โfarโ from the
critical edge.
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
0
2
4
6
8
0.000
0.005
0.010
0.015
0.020
0.025
0.030
0.035
Harish Chintakunta1
, Mustafa หIlhan Akbaยธs2
FPU & ERAU
Cooperative autonomous navigation of emergency vehicles
10. Problem context Mapping onto graphs Graph spectral properties Problem formalization Results
Putting things together
Construct a graph G = (V, E) with discretized points in the
space as vertices, and edges between vertices in a
geometrical neighborhood.
ฮ : con๏ฌguration of the obstacles.
Assign weights wฮ : E โ [0, 1], where wฮ(e) is a function
of distances between the edge and all other obstacles.
Lฮ((ui, uj)) =
โwฮ((ui, uj)), ui = uj
uk =ui
wฮ((ui, uk )), ui = uj
The second eigenvalue ฮป2(Lฮ) will serve as a good
measure for connectivity.
Harish Chintakunta1
, Mustafa หIlhan Akbaยธs2
FPU & ERAU
Cooperative autonomous navigation of emergency vehicles
11. Problem context Mapping onto graphs Graph spectral properties Problem formalization Results
Framing as an optimization problem
EV
SV
0 1 2 3 4 5 6 7 8
0.00
0.25
0.50
0.75
1.00
1.25
1.50
1.75
2.00
0
1
2
3
4
5
6
7
8
0 1 2 3 4 5 6 7 8 0.000.250.500.751.001.251.501.752.00
0
1
2
3
4
5
6
7
8
Harish Chintakunta1
, Mustafa หIlhan Akbaยธs2
FPU & ERAU
Cooperative autonomous navigation of emergency vehicles
12. Problem context Mapping onto graphs Graph spectral properties Problem formalization Results
Solution to optimization problem
1 The acceleration functions of the surrounding vehicles are
optimized to provide a safe path to the EV.
2 Providing a safe path translates to
โMake a stronger connected componentโ
3 The above objective is achieved by:
{aโ
i (t)} = arg max{ai(t)} ฮป2(Lฮ)
s.t. Safety and boundary constraints on ฮ
Harish Chintakunta1
, Mustafa หIlhan Akbaยธs2
FPU & ERAU
Cooperative autonomous navigation of emergency vehicles
14. Problem context Mapping onto graphs Graph spectral properties Problem formalization Results
Conclusions
1 The โsafetyโ of the situation with respect to navigation of
EV in captured by ฮป2(L) of a suitably constructed graph.
2 The con๏ฌguration of the surrounding vehicles is altered to
improve the safety measure.
3 The alteration of the surrounding vehicle con๏ฌguration is
achieved by moving them along the gradient direction of
the safety measure.
Harish Chintakunta1
, Mustafa หIlhan Akbaยธs2
FPU & ERAU
Cooperative autonomous navigation of emergency vehicles
15. Problem context Mapping onto graphs Graph spectral properties Problem formalization Results
Future directions
1 Distributed algorithms to compute the gradient of the
con๏ฌguration.
2 Statistical analysis in a realistic traf๏ฌc situation.
3 Deeper understanding of the theoretical aspects of Fiedler
vector in the context of optimal cooperative driving.
Harish Chintakunta1
, Mustafa หIlhan Akbaยธs2
FPU & ERAU
Cooperative autonomous navigation of emergency vehicles
16. Problem context Mapping onto graphs Graph spectral properties Problem formalization Results
Thank you!!
Harish Chintakunta1
, Mustafa หIlhan Akbaยธs2
FPU & ERAU
Cooperative autonomous navigation of emergency vehicles