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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
813-974-0778