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INTEGRATED VEHICLE DYNAMICS CONTROL IN AUTONOMOUS VEHICLES:
A REAL-TIME MPC APPROACH
Jahan Asgari, H. Eric Tseng, Davor HrovatFord Research Laboratory, Dearborn, USA
Paolo Falcone, Francesco Borrelli, Luigi GlielmoUniversità degli Studi del Sannio, Benevento, Italy
Applicazioni e prospettive del controllo nei veicoli
Politecnico di Milano, 10 maggio 2007
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OutlineIntroduction
Guidance and Navigation Control algorithms
Vehicle Dynamics Control
Control Oriented Vehicle Model
NLMPC (Non Linear MPC) approach
LTV-MPC (Linear Time Varying MPC) approach
Experimental results
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Guidance and Navigation Control Algorithms
Trajectory-ModeGenerator
TrajectoryUpdate
Mode ofOperation
Inner Loop Control
State Estimator
ParameterEstimator
Vehicle andEnvironment
Trajectory-Mode Replanning
Low-Level Control System
Precomputed off-line
u
y
• On board cameras• Infrared• Radars• Gyro, GPS
Yaw, roll, pitch, lateral, longitudinal and vertical stabilization
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Current Autonomous Vehicle Control Design
Trajectory-ModeGenerator
Vehicle andEnvironment
Trajectory-Mode Replanning
Low-Level Control System
Precomputed off-line
Obstacles Detection
and Avoidance Algorithms
PID Controllers Basedon Linear
Vehicle Models
u
yOnline NonlinearOptimization,based on Point-Mass Model andAccelerationConstraints
Used by some DARPA Grand Challenge Vehicles like Alice (Caltech), Stanley (Stanford University)
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To Develop Advanced
Model Based Control Strategies for
Integrated Vehicle Dynamics Control
The Goal
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Classical Vehicle Dynamics Control
Controlling Yaw, Roll, Pitch, Vertical, Lateral and Longitudinal Dynamics via Multiple Input
Active Front Steering (AFS) systemsAnti-lock Braking System (ABS)Electronic Stability Program(ESP)Traction Control (TC)Suspension control systemsActive differential control systems y
lateral
z yaw
pitch
vertical
longitudinal roll
ψ
θx φ Fx FyFz
Front steeringFour brakesEngine torqueActive suspensionsActive differential
Longitudinal, lateral and vertical velocity
Yaw, roll and pitch angles/rates
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Integrated VDC via MPC
ylateral
z yaw
pitch
vertical
longitudinal roll
ψ
θx φ Fx FyFz
Front steeringFour brakesEngine torqueActive suspensionsActive differential
MIMO controller integrating
local and global measurements coming
from GPS, cameras, infrared and radar
...Position and velocity in a global
frame
Enabling path following capabilities
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ScenarioProblem setup:
• Double lane change • Driving on snow/ice, withdifferent entry speeds
Control objective:
Minimize position and orientation errors from reference trajectory by changing the front wheel steering angle and braking at the four wheels
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Challenges
6 DOF modelLongitudinal, lateral, vertical, roll, yaw and pitch dynamics
Highly nonlinear MIMO system with uncertaintiesTire characteristic, trigonometric functions, bilinear nonlinearities
Hard constraintsRate limit in the actuator, vehicle physical limits
Fast sampling timeTypically 20 ms
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Motivations
Autonomous vehicleMilitary vehicles (DARPA Grand Challenge, Urban Challenge, etc…)Futuristic scenario for passenger cars in urban environment
Autonomous vehicle for civilian applicationsStop-and-go, lane assistant, obstacle avoidance, intelligent parking systems, proximity control systems
Improving guidance assistance systemsCombined AFS, ESP and brakes control (Integrated Vehicle Dynamics Control, IVDC)
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OutlineIntroduction
Guidance and Navigation Control algorithms
Vehicle Dynamics Control
Control Oriented Vehicle Model
NLMPC (Non Linear MPC) approach
LTVMPC (Linear Time Varying MPC) approach
Experimental results
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Four Wheels Model
States
XY
xy
ψψ&
&
& Lateral velocityLongitudinal velocityYaw angleYaw rateLateral position (I.F.)Longitudinal position (I.F.)
fα Front slip angle
fcF
flF
Front cornering force
Front longitudinal force
Other variables
fδ Front steering angleInputs
bF FL, FR, RL,RR brakesτ Desired engine torque
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Simplified Driveline
( )( ))()(
)(),()1(tht
tutftξη
ξξ µ
=
=+
Dynamical Model
[ ]rrrlfrfl ,ω,ω,ω,Y,X,ωψ,,ψx,yξ &&&=
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Pacejka Tire model
Semi-empirical model calibrated on
experimental data
),,,( zFsfF µα=
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OutlineIntroduction
Guidance and Navigation Control algorithms
Vehicle Dynamics Control
Control Oriented Vehicle Model
NLMPC (Non Linear MPC) approach
LTVMPC (Linear Time Varying MPC) approach
Experimental results
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Model Predictive Control
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NLMPC Control design
( ) tosubj.
,min UJ tU∆
∆ξOptimization problem
( )( )
,,,
),( ,
,,
,,1,
,
,,
,,,,1
p
tktktk
tk
tktk
tktkstk
Httkuuu
th
uf
+=
∆+=
=
=
=
−
+
K
ξξξη
ξξ µ
Vehicle dynamics
max,,min, ftkf u δδ ≤≤Input constraints
1,, max,,min, −+=∆≤∆≤∆ cftkf Httku KδδConstraints on input changes
( )( ) ∑ ∑=
−
=++ ∆+−=∆
+
p c
tit
H
i
H
iRtitQreftit uUtJ
1
1
0
2,
2
, ,, ηηξCost function
Nonlinear Vehicle Dynamical
(Pacjeka Tire Model)
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The NLMPC controllerNon-linear optimization problem
Non-linear optimization solver is required
High computational burden
Experimental tests are possible at low vehicle speed
Stability guaranteed
Nonlinear MPC in real time used for the first time in fast automotive applications with standard prototyping hardware and off-the-shelf nonlinear solver (sample time 50 ms)
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Lateral and Yaw Stabilization via Active Front Steering (AFS)
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83 Differential Equations
Simulation Enviroment: Carsim
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OutlineIntroduction
Guidance and Navigation Control algorithms
Vehicle Dynamics Control
Vehicle modelling
NLMPC (Non Linear MPC) approach
LTVMPC (Linear Time Varying MPC) approach
Experimental results
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The Gap…
NonlinearMPC
Online/ExplicitLTI/PWA MPC
Problem Domain: System Model, Sampling Time, Computational Resources
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…Filling The Gap
For a MIMO “Fast System”• Nonlinear MPC is not implementable with
current methodologies/ technologies• Linear MPC is unstable / poor performance• Approximated PWA Model is complex (PWA
solution explodes)
Systematic Control Design Procedure with Constraints Fulfillments and Tuning Knobs
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LTV MPC vs PWA
t
Y
PWA Prediction error. Can be large locally. Smaller over the horizon
t
⎥⎦
⎤⎢⎣
⎡
**
**
tt
tt
DCBAY
*t
ReferenceLinear PredictionNonlinear Prediction
LTV Prediction error. Small locally. Large over long horizons
⎥⎦
⎤⎢⎣
⎡
ii
ii
DCBA
ReferencePWA PredictionNonlinear Prediction
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Summary
Nonlinear model. Nonlinear Programming PWA model. Mixed-Integer Programming/ Explicit Solution
Tire Slip
Tire Torqu
e
Piecewise affine approximation
x
)(xf
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11
,,
++
++
kk
kk
DCBA
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22
,,
++
++
kk
kk
DCBA
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33
,,
++
++
kk
kk
DCBA
44
44
,,
++
++
kk
kk
DCBA
LTV model. Quadratic/Linear Programming
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LTV MPC Control design( )( ) ∑ ∑
=
−
=++ +−=∆
+
p c
tit
H
i
H
iRtitQreftit uUtJ
1
1
0
2,
2
, ,, δηηξ
( ) tosubj.
,min UJ tU∆
∆ξ
1,, −−= ttktk uuuδ
max,min uuu tk ≤≤
1,, max,1,min −+=∆≤−≤∆ − ctktk Httkuuuu K
Optimization problem
Linearized Vehicle Dynamical
(Including Pacjeka Tire Model)
Input constraints
Constraints on input changes
Cost function
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The LTV controller
A Quadratic Programming (QP) optimization problem has to be solved
A QP solver is required
Problem solved with small computational effort
Experimental tests even at high speed
Stability not guaranteed!
A Stability Condition has been proposed for such a scheme.
In summary: Stability depends on the prediction mismatch at each time step
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The LTV controller (cont’d)Simulations results at 17 m/s
A linear model is not able to “predict” a slope change of the tire characteristic in the prediction horizon
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Constraints on slip angle
minαmaxα
ptk Httktktk
+=≤≤ K ,, max,min ααα
Controller performs well up
to 21 m/s
State and input constraintThe system is still nonlinear
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The NLMPC controller
The constraints on slip angle are effective both in simulations and in experimental tests
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OutlineIntroduction
Guidance and Navigation Control algorithms
Vehicle Dynamics Control
Vehicle modelling
NLMPC (Non Linear MPC) approach
LTVMPC (Linear Time Varying MPC) approach
Experimental results
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Revi Test Center in Arjeplog, Lapland
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Experimental setup
Sampling time: 50 msThe experimental tests have been done using a dSpace rapid prototyping system equipped with a DS1005 processor boardMain limitation arising from dSpace: source code of the solver has to be availableDifferential GPS, gyros, lateral accelerometersJaguar X-type
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NLMPC AFS controller at 7 m/s (25 Km/h)
The control and the prediction horizons are Hu=3, Hp=7
Problem dimension: 42 nonlinear constraints,12 linear constraints, 3 optimizersNPSOL has been used
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LTVMPC AFS controller at 19 m/s (68.4 Km/h)
The control and the prediction horizons are Hu=10, Hp=25
Problem dimension: 54 linear constraints, 10 optimizersQPDANTZG is used
Bias in yaw angle measurement
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Illustration of Measurement Bias (Single Antenna RT3000)
A
Local X
Bx
y
X
Vehicle coordinate andGlobal coordinate
Illustration of measurement bias takenin Controller B results
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LTVMPC controller at 10 m/s (36 Km/h)
The control and the prediction horizons are Hu=1, Hp=25
Problem dimension: 54 linear constraints, 1 optimizersTailored QP solver is used
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LTVMPC AFS controller at 20m/s (70 Km/h)
The control and the prediction horizons are Hu=10, Hp=25
Problem dimension: 54 linear constraints, 10 optimizersQPDANTZG is used
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LTVMPC AFS controller at 21m/s (75.6 Km/h)
The control and the prediction horizons are Hu=1, Hp=25
Problem dimension: 54 linear constraints, 1 optimizersTailored QP solver is used
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Braking and steering LTVMPC controller at 70 Kph
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Braking and steering LTVMPV controller at 70 Kph (cont’d)
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Braking and steering LTVMPV controller at 70 Kph (cont’d)
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ConclusionsAFS and combined AFS and braking control problems have been presented
Nonlinear MPC controllers have been tested both in simulations and in experimental tests
A complex AFS NLMPC controller has been successfully implemented in real-time and experimentally validated at low speeds
A suboptimal MPC controller based on on-line linearizations has been designed and validated
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Remarks
Sistematic control approach for Integrated VehicleDynamics ControlComputational burden can be decreased through suboptimal schemesStability and performance results have to beprovided for these non standard schemesExperimental setup