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Control of Connected Autonomous Vehicles in Mixed Traffic: Modeling

and Field ExperimentsXiaopeng (Shaw) Li

Associate Professor, Susan A Bracken FellowUniversity of South Florida (USF)

4/25/2019CUTR Transportation Webcast Series

Connected Vehicles• Vehicle connection = Information sharing

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Automated Vehicles• Human drivers → Robot drivers

Connected Automated Vehicles (CAVs)• Enable vehicle trajectory-level control

• Transformation: accommodate human driving → design vehicle trajectories

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Opportunities for CAV Trajectory Design• Individual vehicles in controlled environment → streams of

vehicles on a road segment (may need to consider uncontrolled human drivers)

• Computationally intensive algorithms → real-time scalable models

• Numerical and general data-driven approaches → analytical insights and traffic flow domain sensitive methods

• Simulation → field experiments

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Our Approach on CAV Trajectory Design

Theoretical analysis

Fast heuristic solutions

Accelerated exact algorithms

Validation and demonstration with field experiments

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General Longitudinal Problem• Infrastructure – a road segment• A stream of CAV trajectories 𝑠 (from 1 to I)

• Boundary conditions: e.g., initial location 𝑠 and speed 𝑣 at time 0, final location range [𝑠 , 𝑠 at time 𝑇…

Time

Spac

e

𝑠 , 𝑣 )

[𝑠,𝑠

Veh 1…

Veh 𝑖

Veh 𝐼

…

𝑇0

𝑠

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Physical Limits• Speed 𝑠 𝑡 ∈ 0, 𝑣

• Acceleration 𝑠 𝑡 ∈ 𝑑 , 𝑎

Time

Spac

e

𝑠 , 𝑣 )

[𝑠,𝑠

Veh 1

…

Veh 𝑖

Veh 𝐼

…

𝑇08

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Safety Constraints• Two consecutive trajectories 𝑠 and 𝑠

• Shifted trajectory �̂� 𝑡 𝑠 𝑡 𝑔 ; jam spacing 𝑔 , response time

• Safety constraint:�̂� 𝑡 𝑠 𝑡 0, ∀𝑡

𝑠

𝑔

�̂�

𝑠Consistent with the triangular fundamental diagram

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Objectives• Mobility max ∑ 𝑠 𝑡,

• Driving comfort min ∑ 𝑠 𝑡,

• Fuel consumption min ∑ 𝑒 𝑠 𝑡 , 𝑠 𝑡,

• Safety surrogate

min ∑ 𝑓 𝑠 𝑡 , 𝑠 𝑡 , 𝑠 𝑡 , 𝑠 𝑡 ,

• General form min ∑ 𝐽 𝑠 𝑡

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Problem Formulation

min 𝐽 𝑠 𝑡

s.t.

𝑠 0 𝑠 , 𝑠 0 𝑣 , 𝑠 𝑠 𝑇 𝑠 , ∀𝑖

𝑑 𝑠 𝑡 𝑎 ∀𝑖, 𝑡

0 𝑠 𝑡 𝑣 ∀𝑖, 𝑡

𝑔 𝑠 𝑡 𝜏 𝑠 𝑡 , ∀𝑖 ∈ ℐ\ 1 , 𝑡

Complex nonlinear program with differential constraints

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Theoretical Analysis• Shift coordinates to eliminate spacing and response time

𝑦 𝑡 𝑠 𝑡 𝑖 1 𝜏 𝑖 1 𝑔

𝑠

𝑠

𝑠

𝑦 𝑡)

𝑔

𝑦

2𝑔

𝑦 𝑡)

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Theoretical Analysis• Safety constraints reduce to

𝑦 𝑡 𝑦 𝑡

𝑠

𝑠

𝑠

𝑦 𝑡)

𝑔

𝑦

2𝑔

𝑦 𝑡)

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Local Feasibility Analysis• Individual bounds to 𝑦 without considering safety

– Upper bound �̅� : Acceleration maximally until having to decelerate maximally to meet the boundary conditions

– Lower bound 𝑥 : Deceleration maximally until having to accelerate maximally to meet the boundary conditions

– Feasible region (without safety) - �̅� 𝑦 𝑥 - Second-order space-time prism

Time

Spac

e

𝑇

𝑠𝑠

𝑥

�̅�

𝑠 , 𝑣 )

Bounds: analytical piecewise  polynomial functions  

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Global Feasibility Theorems• Global upper bound 𝑦 to 𝑦 : smoothed lower envelope to �̅� , ⋯ , �̅�• Global lower bound 𝑦 to 𝑦 : smoothed upper envelop to 𝑥 , ⋯ , 𝑥

• Feasible region (considering all constraints) 𝑦 𝑦 𝑦

• Easy to convert 𝑦 ,𝑦 in the original coordinates as �̅� ,𝑠

Time𝑇

Bounds: analytical piecewise polynomial functions  

�̅�

�̅�

…

𝑥

…

𝑥

𝑦

𝑦

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Special Optimal Solution• Theorem: For mobility objective alone max ∑ 𝑠 𝑡, , the

optimal trajectory solution is �̅�

Time𝑇

�̅�

�̅�

……

�̅�

Reference: Li, L. & Li, X. (2019). “Parsimonious trajectory design of connected automated traffic.” Transportation Research Part B, 119, 1‐21.  16

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Algorithmic Implications• For problems with more

general objectives

• Exact mathematical programming on a time space network (after discretization)

• Implications of theoretical analysis: Reduce state space of decision variables to improve the solution efficiency

min 𝐽 𝑠

s.t.

𝑠 𝑠 , 𝑠 𝑣 ,

𝑠 𝑠 𝑠 , ∀𝑖

  𝑑 𝑠 𝑎 ∀𝑖, 𝑡

0 𝑠 𝑣 ∀𝑖, 𝑡

𝑔 𝑠 𝑠 , ∀𝑖 ∈ ℐ\ 1 , 𝑡

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Algorithmic Implications• Construction of fast heuristic algorithms

• Find a smooth piece-wise polynomial trajectories between bounds 𝑠 , �̅�

Reference: Li, X., Ghiasi, A., Xu, Z. & Qu, X. (2018). “A piecewise trajectory optimization model for connected automated vehicles: Exact optimization algorithm and queue propagation analysis.” Transportation Research Part B, 118: 429‐456.Zhou, F., Li, X. & Ma, J. (2017). “Parsimonious shooting heuristic for trajectory design of connected automated traffic part I: Theoretical analysis with generalized time geography.” Transportation Research Part B, 95, 394‐420.

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Algorithm Performance Comparison• Exact method vs. fast heuristics on fuel consumption objectives

Parameter values Exact obj Heuristic obj gapExact 

time/sec

Heuristic

time/sec

Default 12133.7 12828.3 5.4% 461.7 0.894

N=10 6096.9 6462.7 5.7% 106.4 0.885

N=30 18555.7 19252.8 3.6% 821.2 0.879

L=500 7572.9 8596.8 11.9% 235.2 0.923

L=1000 17430.0 17709.4 1.6% 608.72 0.926

rin=0.2, rout=0.6 17152.1 17272.0 0.7% 238.9 0.883

rin=0.6, rout=0.2 12197.6 12573.2 3.0% 1051.0 0.905

=10 s 14182.2 14527.8 2.4% 259.4 0.885

=30 s 11453.9 12106.6 5.4% 556.1 0.876

𝜎 2 16+/‐2 13603.1 14776.0 7.9% 419.1 1.631

𝜎 4 16+/‐4 15121.9 17397.0 13.1% 387.5 1.293

Average 5.53% 467.7 0.998

References: 19‐04982_D ‐ Trajectory Optimization for a Connected Automated Traffic Stream: Comparison Between an Exact Model and Fast Heuristics19‐04445_B ‐ A Joint Trajectory and Signal Optimization Model for Connected Automated Vehicles

FS ( / )VSPSHC kJ tonETO( / )VSPC kJ tonVSP SSFS ( / )VSPSHC kJ tonETO( / )VSPC kJ tonVSP SSFS ( / )VSPSHC kJ tonETO( / )VSPC kJ tonVSP SSFS ( / )VSPSHC kJ tonETO( / )VSPC kJ tonVSP SS

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Field Experiments• Theoretical analysis & algorithm developments provide

real-time methods for design smooth trajectories for real-world CAV control

• Need to be integrated with real-world vehicle settings and human driving behavior

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• 2.4 km (1.5 miles) track, Chang’an University, Xi’an China• CAVs, instrumented vehicles, road side devices, signals

Facilities Provided by Partner Institute

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Test 1: HV following AV• Ten drivers • Constant & dynamic lead vehicle speed • Speed range: 0-60km/h

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Constant Speed Experiments• Space time headway distribution

Time headway

Space hea

dway

Time

Space

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Constant speed results

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Dynamic Speed Experiments• Car-following dynamics

Time

Space

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Following HV vs. AV

𝒩: Set of all test drivers𝒪: Set of test vehicles order.L refers to the lead vehicle and F refers to the following vehicleℳ: Set of the lead vehicle’s driving mode.I indicates autonomous mode and H means human driven mode𝑎 𝑡 : Vehicle acceleration𝑝 𝑡 : Vehicle location𝑣 𝑡 : Vehicle speed𝑠 : Safety distance𝑘 , 𝛼 , 𝜆 : Factors of the FVD model𝑅 𝑡 : Residual error

𝑎 𝑡

𝑘 𝛼 𝑝 𝑡 𝜏 𝑝 𝑡 𝜏 𝑠 𝑣 𝑡 𝜏 𝜆 𝑣 𝑡 𝜏 𝑣 𝑡 𝜏

𝑅 𝑡 , ∀𝑛 ∈ 𝒩, 𝑡 ∈ 𝒯

𝑎 𝑡

𝑘 𝛼 𝑝 𝑡 𝜏 𝑝 𝑡 𝜏 𝑠 𝑣 𝑡 𝜏 𝜆 𝑣 𝑡 𝜏 𝑣 𝑡 𝜏

𝑅 𝑡 , ∀𝑛 ∈ 𝒩, 𝑡 ∈ 𝒯

Trajectory, speed and acceleration data are used to calibrate the FVD model for each driver in each following mode with linear regression.

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Open-Access Results• Data: https://github.com/sgzzgit/Field-Experiment-Data

• Manuscript: https://www.researchgate.net/publication/329772815_Field_Experiments_on_Longitudinal_Characteristics_of_Human_Driver_Behavior_Following_an_Autonomous_Vehicle

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Test 2: AV lane change in mixed traffic

1

2 4

3

NO. 3 : AV; Others HV

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Step 1：Detect the vehicles around with the LiDAR (velodyne). Calculate the relative distance and speed of vehicles.AV follows both vehs # 1 and 2 with the ACC model (developed by PATH)

𝑎 : Target acceleration of AV when following vehicle 𝑖, 𝑖 ∈ 1,2𝑎 : Acceleration input of the AV𝑝 : ith vehicle position. 𝑖 ∈ 1,2,3,4𝑣 : ith vehicle speed. 𝑖 ∈ 1,2,3,4𝑘 , 𝑘 : Parameters of the model

𝑎 𝑘 𝑝 𝑝 𝜏𝑣 𝑘 𝑣 𝑣𝑎 𝑘 𝑝 𝑝 𝜏𝑣 𝑘 𝑣 𝑣

𝑎 min 𝑎 , 𝑎

Longitudinal - Front Vehicles

1

2

34

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Step 2：Check whether it is safe to execute lane changing – not causing too dramatic deceleration for veh # 4.

𝑏: Maximum comfortable deceleration

Longitudinal - Rear Vehicle

1

2

3

4

𝑎 𝑏 & 𝑎 𝑏𝑎 𝑏

& 𝑎 𝑏

𝑎 𝑏 𝑎 𝑏

Yes

Yes

Lane changing

Car following

No

No

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Step 3：Lane change pathgeneration based on the sine function.

Lane-Change Path Generation

1

2

3

4

𝐿

𝑌𝑌 𝑥𝑌2𝜋

2𝜋𝐿

𝑥 𝑠𝑖𝑛2𝜋𝐿

𝑥 , 𝑥 ∈ 0, 𝐿

𝐿 : AV distance from current pos to the target pos in the adjacent laneDynamically adjusted based on the vehicle’s real time speed to make the lateral acceleration falls in a comfortable range

𝑌 : Distance between two adjacent lanes

Longitudinal control always activatedIf the longitudinal safety check fails anytime before CAV passes the lane marker, the lane changing aborts

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Step 4：Path following based on the pure-pursuit algorithm. Use the Model–view–controller architecture to ensure real-time control with minimum lag

Lane-Change Path Following

𝛿 𝑡 tan2𝐿𝑠𝑖𝑛 𝛼 𝑡

𝑘𝑣 𝑡

𝛿 𝑡 ∶ Target vehicle steering angle𝑣 𝑡 ∶ Autonomous vehicle speedL ∶ 2.66 𝑚k ∶ 0.5

𝛿 𝑡 will be sent to the CAN BUS

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Field Experiments

No congestion AV lane congested

Target lane congested Lane changing aborted

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Comparison between HV and AVFindings: AV has milder steering angle AV has smoother speed

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Test 3: CAV Control in Mixed Traffic at a Signalized Intersection

V2IV2I

1 32 4 5

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Field Experiments1 downstream HV 2 downstream HVs

3 downstream HVs

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USF CAV Testbed

• Hardware (Thanks to USF R&I, CoE): – Vehicle - Lincoln MKZ Hybrid

(from Parks Lincoln of Tampa)– Sensors (from AutonomouStuff) – 2

Velodyne 16-beam Lidars, one Delphi mili-meter Radar Kit, one HD NovAtel Navigation unit, one MobileEye Development Kit, one FLIR Grey Point Camera,

– Computing – Spectra industry computer

– Control – Customized by-wire control

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USF CAV Testbed

• Units – Savari Dedicated Short Range Communication – 4 Road Side Units (RSU)– 5 Onboard Units (OBU)

• Development –– Portable OBUs – Suitcase kits. Easy to

integrate with any existing vehicle– Portable RSUs – Movable tripod-like RSU.

Simulate a signal network

• Research – Sensing, computing and control of

connected autonomous vehicle (CAV)– Implications to traffic and infrastructure

management – Security

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Acknowledgements• Collaborators –

– Yu Wang (USF)

– Saied Soleimaniamiri (USF)

– Zhen Wang (Chang’an University)

– Zhigang Xu (Chang’an University)

– Xiangmo Zhao (Changan University)

• Funding – NSF CMMI # 1558887; USF funds; Chang’anUnviersity Funds

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ThanksQ & A

Xiaopeng (Shaw) Li

xiaopengli@usf.edu

813-974-0778