44
Bifurcation analysis of vehicle and vehicle+driver models Fabio Della Rossa Dipartimento di Elettronica, Informazione e Bioingegneria
Bifurcation analysis of vehicle and vehicle+driver models
Fabio Della Rossa
Dipartimento di Elettronica, Informazione e Bioingegneria
Bifurcation analysis of an automobilemodel negotiating a curve Rinaldi, Piccardi, Dercole, Gragnani, Miari, Colombo, Della Rossa
DEIB – Dinamica Sistemi Complessi
Who we are…
SergioRinaldiEm. prof.
CarloPiccardiFull prof.
FabioDercoleAss. prof.
AlessandraGragnani
MassimoMiari
AlessandroColombo
FabioDella Rossa
Bifurcation analysis of an automobilemodel negotiating a curve Rinaldi, Piccardi, Dercole, Gragnani, Miari, Colombo, Della Rossa
DEIB – Dinamica Sistemi Complessi
Theory Bifurcation studies in smooth and
nonsmooth systems
RelatoreNote di presentazione
Bifurcation analysis of an automobilemodel negotiating a curve Rinaldi, Piccardi, Dercole, Gragnani, Miari, Colombo, Della Rossa
DEIB – Dinamica Sistemi Complessi
Theory Bifurcation studies in smooth and
nonsmooth systems
Nonlinear control
RelatoreNote di presentazione
Bifurcation analysis of an automobilemodel negotiating a curve Rinaldi, Piccardi, Dercole, Gragnani, Miari, Colombo, Della Rossa
DEIB – Dinamica Sistemi Complessi
Theory Bifurcation studies in smooth and
nonsmooth systems
Nonlinear control
Complex networks analysis
RelatoreNote di presentazione
Bifurcation analysis of an automobilemodel negotiating a curve Rinaldi, Piccardi, Dercole, Gragnani, Miari, Colombo, Della Rossa
DEIB – Dinamica Sistemi Complessi
Theory Bifurcation studies in smooth and
nonsmooth systems
Nonlinear control
Complex networks analysis
Intelligent transportation systems
RelatoreNote di presentazione
Bifurcation analysis of an automobilemodel negotiating a curve Rinaldi, Piccardi, Dercole, Gragnani, Miari, Colombo, Della Rossa
DEIB – Dinamica Sistemi Complessi
Theory Bifurcation studies in smooth and
nonsmooth systems
Nonlinear control
Complex networks analysis
Intelligent transportation systems
Innovation and competition processes
RelatoreNote di presentazione
Bifurcation analysis of an automobilemodel negotiating a curve Rinaldi, Piccardi, Dercole, Gragnani, Miari, Colombo, Della Rossa
DEIB – Dinamica Sistemi Complessi
Theory Bifurcation studies in smooth and
nonsmooth systems
Nonlinear control
Complex networks analysis
Intelligent transportation systems
Innovation and competition processes
Evolutionary game theory
RelatoreNote di presentazione
Bifurcation analysis of an automobilemodel negotiating a curve Rinaldi, Piccardi, Dercole, Gragnani, Miari, Colombo, Della Rossa
DEIB – Dinamica Sistemi Complessi
Applications
Interspecific competition
Pred
ator
atta
ck ra
te
Stability of homogeneus solutions in spatial distributed ecological models
RelatoreNote di presentazione
Bifurcation analysis of an automobilemodel negotiating a curve Rinaldi, Piccardi, Dercole, Gragnani, Miari, Colombo, Della Rossa
DEIB – Dinamica Sistemi Complessi
Applications Stability of homogeneus solutions in spatial
distributed ecological models Modeling and analysis of human behavioral
systems
RelatoreNote di presentazione
Bifurcation analysis of an automobilemodel negotiating a curve Rinaldi, Piccardi, Dercole, Gragnani, Miari, Colombo, Della Rossa
DEIB – Dinamica Sistemi Complessi
Applications Stability of homogeneus solutions in spatial
distributed ecological models Modeling and analysis of human behavioral
systems Evolutionary systems, fisheries
susteinability, and evolutionary branching
RelatoreNote di presentazione
Bifurcation analysis of an automobilemodel negotiating a curve Rinaldi, Piccardi, Dercole, Gragnani, Miari, Colombo, Della Rossa
DEIB – Dinamica Sistemi Complessi
Applications Stability of homogeneus solutions in spatial
distributed ecological models Modeling and analysis of human behavioral
systems Evolutionary systems, fisheries
susteinability, and evolutionary branching Attitude stability analisis of a magnetically
controlled spacecraft
RelatoreNote di presentazione
Bifurcation analysis of an automobilemodel negotiating a curve Rinaldi, Piccardi, Dercole, Gragnani, Miari, Colombo, Della Rossa
DEIB – Dinamica Sistemi Complessi
Applications Stability of homogeneus solutions in spatial
distributed ecological models Modeling and analysis of human behavioral
systems Evolutionary systems, fisheries
susteinability, and evolutionary branching Attitude stability analisis of a magnetically
controlled spacecraft Stability analysis of DC distribution systems
with a constant power load (boats)
RelatoreNote di presentazione
Bifurcation analysis of an automobilemodel negotiating a curve Rinaldi, Piccardi, Dercole, Gragnani, Miari, Colombo, Della Rossa
DEIB – Dinamica Sistemi Complessi
Applications Stability of homogeneus solutions in spatial
distributed ecological models Modeling and analysis of human behavioral
systems Evolutionary systems, fisheries
susteinability, and evolutionary branching Attitude stability analisis of a magnetically
controlled spacecraft Stability analysis of DC distribution systems
with a constant power load (boats) Non-deterministic chaos in electronic
circuits
RelatoreNote di presentazione
Bifurcation analysis of an automobilemodel negotiating a curve Rinaldi, Piccardi, Dercole, Gragnani, Miari, Colombo, Della Rossa
DEIB – Dinamica Sistemi Complessi
Applications Stability of homogeneus solutions in spatial
distributed ecological models Modeling and analysis of human behavioral
systems Evolutionary systems, fisheries
susteinability, and evolutionary branching Attitude stability analisis of a magnetically
controlled spacecraft Stability analysis of DC distribution systems
with a constant power load (boats) Non-deterministic chaos in electronic
circuits Bifurcation analysis of neural mass models
(epilepsy)
RelatoreNote di presentazione
Bifurcation analysis of an automobilemodel negotiating a curve Rinaldi, Piccardi, Dercole, Gragnani, Miari, Colombo, Della Rossa
DEIB – Dinamica Sistemi Complessi
Applications Stability of homogeneus solutions in spatial
distributed ecological models Modeling and analysis of human behavioral
systems Evolutionary systems, fisheries
susteinability, and evolutionary branching Attitude stability analisis of a magnetically
controlled spacecraft Stability analysis of DC distribution systems
with a constant power load (boats) Non-deterministic chaos in electronic
circuits Bifurcation analysis of neural mass models
(epilepsy) Analysis and control of vehicular traffic
RelatoreNote di presentazione
Bifurcation analysis of an automobilemodel negotiating a curve Rinaldi, Piccardi, Dercole, Gragnani, Miari, Colombo, Della Rossa
DEIB – Dinamica Sistemi Complessi
Applications Stability of homogeneus solutions in spatial
distributed ecological models Modeling and analysis of human behavioral
systems Evolutionary systems, fisheries
susteinability, and evolutionary branching Attitude stability analisis of a magnetically
controlled spacecraft Stability analysis of DC distribution systems
with a constant power load (boats) Non-deterministic chaos in electronic
circuits Bifurcation analysis of neural mass models
(epilepsy) Analysis and control of vehicular traffic Vehicle lateral dynamics
RelatoreNote di presentazione
Bifurcation analysis of an automobilemodel negotiating a curve Rinaldi, Piccardi, Dercole, Gragnani, Miari, Colombo, Della Rossa
DEIB – Dinamica Sistemi Complessi
Applications Stability of homogeneus solutions in spatial
distributed ecological models Modeling and analysis of human behavioral
systems Evolutionary systems, fisheries
susteinability, and evolutionary branching Attitude stability analisis of a magnetically
controlled spacecraft Stability analysis of DC distribution systems
with a constant power load (boats) Non-deterministic chaos in electronic
circuits Bifurcation analysis of neural mass models
(epilepsy) Analysis and control of vehicular traffic Vehicle lateral dynamics
Theory Bifurcation studies in smooth and
nonsmooth systems
Nonlinear control
Complex networks analysis
Intelligent transportation systems
Innovation and competition processes
Evolutionary game theory
RelatoreNote di presentazioneCONCLUDERE CON UNA SLIDE CON TUTTE LE COSE, E DIRE DI CHE COSA HO INTENZIONE DI PARLARE OGGI
Bifurcation analysis of an automobilemodel negotiating a curve Rinaldi, Piccardi, Dercole, Gragnani, Miari, Colombo, Della Rossa
DEIB – Dinamica Sistemi Complessi
THE HUMAN-IN-THE-LOOP SYSTEM
• Driver models and vehicle-driver coupling• Model validation• Bifurcation analysis of the vehicle and driver model• Results comparison• An extension and further research directions
THE AUTOMOBILE MODEL AND ITS BA
• The single track model• Model validation• Bifurcation analysis of the vehicle model• Understeering vs. Oversteering vehicles
RelatoreNote di presentazioneBefore beginning some remarks on the presentation:Two parameter bifurcation analysis of dynamical systems: is the main theme of the thesis, and I will begin this talk explaining what is it and why is it important.Since the theme of the thesis is quite complex and not so popular, I will spend half of my presentation on the introduction.So, lets start.
Bifurcation analysis of an automobilemodel negotiating a curve Rinaldi, Piccardi, Dercole, Gragnani, Miari, Colombo, Della Rossa
DEIB – Dinamica Sistemi Complessi
The single track model
SIMPLIFYING HYPOTHESES • the forward speed u is constant;• the centre of gravity lies at the ground level;• the vehicle body is modelled referring to its longitudinal axis;• the resultant of the forces acting at the front and rear axles are applied at the centres of theaxles;• the slip angles αi, i = 1, 2, and the steering angle δ (Figure 1) are small and• no longitudinal forces are acting at the wheels.
Where the rear and front slip angles can be obtained as:
𝑚𝑚 (�̈�𝑦 + 𝑢𝑢 �̇�𝜃) = 𝐹𝐹𝑦𝑦1(𝛼𝛼1) + 𝐹𝐹𝑦𝑦2(𝛼𝛼2)𝐼𝐼𝑧𝑧 �̈�𝜃 = 𝐹𝐹𝑦𝑦1(𝛼𝛼1)𝑎𝑎 − 𝐹𝐹𝑦𝑦2(𝛼𝛼2)𝑏𝑏
𝑦𝑦𝜃𝜃
𝛼𝛼2 = −�̇�𝑦 − 𝑏𝑏�̇�𝜃
u, 𝛿𝛿 − 𝛼𝛼1 =
�̇�𝑦 + 𝑎𝑎 �̇�𝜃𝑢𝑢
2 D.O.F. MODEL IN �̇�𝑦 and �̇�𝜃
Bifurcation analysis of an automobilemodel negotiating a curve Rinaldi, Piccardi, Dercole, Gragnani, Miari, Colombo, Della Rossa
DEIB – Dinamica Sistemi Complessi
The single track model – non-linearities
Where the rear and front slip angles can be obtained as:
Tyre characteristics
UN
1ve
hicl
e
𝑚𝑚 (�̈�𝑦 + 𝑢𝑢 �̇�𝜃) = 𝐹𝐹𝑦𝑦1(𝛼𝛼1) + 𝐹𝐹𝑦𝑦2(𝛼𝛼2)𝐼𝐼𝑧𝑧 �̈�𝜃 = 𝐹𝐹𝑦𝑦1(𝛼𝛼1)𝑎𝑎 − 𝐹𝐹𝑦𝑦2(𝛼𝛼2)𝑏𝑏
𝛼𝛼2 = −�̇�𝑦 − 𝑏𝑏�̇�𝜃
u, 𝛿𝛿 − 𝛼𝛼1 =
�̇�𝑦 + 𝑎𝑎 �̇�𝜃𝑢𝑢
𝑦𝑦𝜃𝜃
Bifurcation analysis of an automobilemodel negotiating a curve Rinaldi, Piccardi, Dercole, Gragnani, Miari, Colombo, Della Rossa
DEIB – Dinamica Sistemi Complessi
Model validation
Bifurcation analysis of an automobilemodel negotiating a curve Rinaldi, Piccardi, Dercole, Gragnani, Miari, Colombo, Della Rossa
DEIB – Dinamica Sistemi Complessi
Try it with PPlane
Bifurcation analysis of an automobilemodel negotiating a curve Rinaldi, Piccardi, Dercole, Gragnani, Miari, Colombo, Della Rossa
DEIB – Dinamica Sistemi Complessi
The single track model – the UN1 case
Bifurcation analysis of an automobilemodel negotiating a curve Rinaldi, Piccardi, Dercole, Gragnani, Miari, Colombo, Della Rossa
DEIB – Dinamica Sistemi Complessi
The single track model – the UN1 case
Bifurcation analysis of an automobilemodel negotiating a curve Rinaldi, Piccardi, Dercole, Gragnani, Miari, Colombo, Della Rossa
DEIB – Dinamica Sistemi Complessi
Try it with PPlane
Bifurcation analysis of an automobilemodel negotiating a curve Rinaldi, Piccardi, Dercole, Gragnani, Miari, Colombo, Della Rossa
DEIB – Dinamica Sistemi Complessi
Bifurcation analysis – the UN1 case
Bifurcation analysis of an automobilemodel negotiating a curve Rinaldi, Piccardi, Dercole, Gragnani, Miari, Colombo, Della Rossa
DEIB – Dinamica Sistemi Complessi
Bifurcation analysis– the UN1 case
Bifurcation analysis of an automobilemodel negotiating a curve Rinaldi, Piccardi, Dercole, Gragnani, Miari, Colombo, Della Rossa
DEIB – Dinamica Sistemi Complessi
The single track model – non-linearities
Where the rear and front slip angles can be obtained as:
Tyre characteristics
OV 0
ve
hicl
e
Bifurcation analysis of an automobilemodel negotiating a curve Rinaldi, Piccardi, Dercole, Gragnani, Miari, Colombo, Della Rossa
DEIB – Dinamica Sistemi Complessi
Try it with PPlane
Bifurcation analysis of an automobilemodel negotiating a curve Rinaldi, Piccardi, Dercole, Gragnani, Miari, Colombo, Della Rossa
DEIB – Dinamica Sistemi Complessi
Bifurcation analysis – OV0 case
δ
Bifurcation analysis of an automobilemodel negotiating a curve Rinaldi, Piccardi, Dercole, Gragnani, Miari, Colombo, Della Rossa
DEIB – Dinamica Sistemi Complessi
The single track model – non-linearities
Where the rear and front slip angles can be obtained as:
Tyre characteristics
OV 2
ve
hicl
e
Bifurcation analysis of an automobilemodel negotiating a curve Rinaldi, Piccardi, Dercole, Gragnani, Miari, Colombo, Della Rossa
DEIB – Dinamica Sistemi Complessi
Bifurcation analysis – OV2 case
Bifurcation analysis of an automobilemodel negotiating a curve Rinaldi, Piccardi, Dercole, Gragnani, Miari, Colombo, Della Rossa
DEIB – Dinamica Sistemi Complessi
Bifurcation analysis – some conclusions
• Propose a method that, for each car, shows in the parameter space the safety boundaries
• Find some behaviour that was already been seen in experiments but was never been founded in simulations
• Classify all the possible behaviours of this system and understand which are the causes that make vehicle unstable (or not robust)
Bifurcation analysis of an automobilemodel negotiating a curve Rinaldi, Piccardi, Dercole, Gragnani, Miari, Colombo, Della Rossa
DEIB – Dinamica Sistemi Complessi
THE HUMAN-IN-THE-LOOP SYSTEM
• Driver models and vehicle-driver coupling• Model validation• Bifurcation analysis of the vehicle and driver model• Results comparison• An extension and further research directions
THE AUTOMOBILE MODEL AND ITS BA
• The single track model• Model validation• Bifurcation analysis of the vehicle model• Understeering vs. Oversteering vehicles
RelatoreNote di presentazioneBefore beginning some remarks on the presentation:Two parameter bifurcation analysis of dynamical systems: is the main theme of the thesis, and I will begin this talk explaining what is it and why is it important.Since the theme of the thesis is quite complex and not so popular, I will spend half of my presentation on the introduction.So, lets start.
Bifurcation analysis of an automobilemodel negotiating a curve Rinaldi, Piccardi, Dercole, Gragnani, Miari, Colombo, Della Rossa
DEIB – Dinamica Sistemi Complessi
Real trajectory
(�̇�𝑦, �̇�𝜃)steer angle
𝛿𝛿
Vehicle+Driver modelVEHICLE MODEL
Desired trajectory
(𝑋𝑋𝑅𝑅 ,𝑌𝑌𝑅𝑅) DRIVERerror
SIMPLIFYING HYPOTHESES • the forward speed u is constant;• the centre of gravity lies at the ground level;• the vehicle body is modelled referring to its longitudinal axis;• the resultant of the forces acting at the front and rear axles are applied at the centres of theaxles;• the slip angles αi, i = 1, 2, and the steering angle δ (Figure 1) are small and• no longitudinal forces are acting at the wheels.
𝑋𝑋
𝑌𝑌
𝑦𝑦𝜃𝜃
Bifurcation analysis of an automobilemodel negotiating a curve Rinaldi, Piccardi, Dercole, Gragnani, Miari, Colombo, Della Rossa
DEIB – Dinamica Sistemi Complessi
error Real trajectory
(𝑋𝑋,𝑌𝑌)steer angle
𝛿𝛿PREDICTION CONTROL
𝑋𝑋
𝑌𝑌
Desired trajectory
(𝑋𝑋𝑅𝑅 ,𝑌𝑌𝑅𝑅)
predictederror VEHICLE
MODEL
𝒆𝒆 = 𝑋𝑋𝑅𝑅𝑌𝑌𝑅𝑅− 𝑋𝑋𝑌𝑌 + 𝐿𝐿
cos 𝜃𝜃sin𝜃𝜃
𝒆𝒆
𝛿𝛿 𝑡𝑡 + 𝜏𝜏𝑟𝑟 = ℎ 𝒆𝒆(𝑡𝑡)
𝑋𝑋
𝑌𝑌
SIMPLIFYING HYPOTHESES • …• the desired trajectory is going straight at 𝑌𝑌𝑅𝑅 = 0;• frequency of human steering < 3 Hz;
𝛿𝛿 𝑡𝑡 + 𝜏𝜏𝑟𝑟�̇�𝛿 𝑡𝑡 = ℎ 𝒆𝒆(𝑡𝑡)
𝑒𝑒 = 𝑌𝑌𝑅𝑅 − (𝑌𝑌 + 𝐿𝐿 sin𝜃𝜃)
𝑚𝑚 (�̈�𝑌 + 𝑢𝑢 �̇�𝜃) = 𝐹𝐹𝑦𝑦1(𝛼𝛼1) + 𝐹𝐹𝑦𝑦2(𝛼𝛼2)𝐼𝐼𝑧𝑧 �̈�𝜃 = 𝐹𝐹𝑦𝑦1(𝛼𝛼1)𝑎𝑎 − 𝐹𝐹𝑦𝑦2(𝛼𝛼2)𝑏𝑏
𝛼𝛼2 = 𝜃𝜃 −�̇�𝑌 − 𝑏𝑏�̇�𝜃
u, 𝜃𝜃 + 𝛿𝛿 − 𝛼𝛼1 =
�̇�𝑌 + 𝑎𝑎 �̇�𝜃𝑢𝑢
𝑦𝑦𝜃𝜃
5 D.O.F. MODEL IN (𝑌𝑌, �̇�𝑌,𝜃𝜃, �̇�𝜃,𝛿𝛿)
Vehicle+Driver model
Bifurcation analysis of an automobilemodel negotiating a curve Rinaldi, Piccardi, Dercole, Gragnani, Miari, Colombo, Della Rossa
DEIB – Dinamica Sistemi Complessi
Model validation – Kick plate
Bifurcation analysis of an automobilemodel negotiating a curve Rinaldi, Piccardi, Dercole, Gragnani, Miari, Colombo, Della Rossa
DEIB – Dinamica Sistemi Complessi
Result comparison: moving straight ahead at different speedUN case
u [m/s]𝜃𝜃 [rad]
𝑌𝑌 [m]
time [s]
𝑌𝑌 [m]• Supercritical Hopf bifurcation• Multiple tangent of limit cycle bifurcation
Bifurcation analysis of an automobilemodel negotiating a curve Rinaldi, Piccardi, Dercole, Gragnani, Miari, Colombo, Della Rossa
DEIB – Dinamica Sistemi Complessi
Result comparison: moving straight ahead at different speedOV case
u [m/s]
𝜃𝜃 [rad]
𝑌𝑌 [m]
• Subcritical Hopf bifurcation
Bifurcation analysis of an automobilemodel negotiating a curve Rinaldi, Piccardi, Dercole, Gragnani, Miari, Colombo, Della Rossa
DEIB – Dinamica Sistemi Complessi
Comparison between UN and OV vehiclesL [m]
𝑢𝑢 [m/s]
𝐿𝐿: length of prevision (driver ability in prevision)𝑢𝑢: forward velocity
Bifurcation analysis of an automobilemodel negotiating a curve Rinaldi, Piccardi, Dercole, Gragnani, Miari, Colombo, Della Rossa
DEIB – Dinamica Sistemi Complessi
error Real trajectory
(𝑋𝑋,𝑌𝑌)steer angle
𝛿𝛿PREDICTION CONTROL
𝑋𝑋
𝑌𝑌
Desired trajectory
(𝑋𝑋𝑅𝑅 ,𝑌𝑌𝑅𝑅)
predictederror VEHICLE
MODEL
𝒆𝒆 = 𝑋𝑋𝑅𝑅𝑌𝑌𝑅𝑅− 𝑋𝑋𝑌𝑌 + 𝐿𝐿
cos𝜃𝜃sin𝜃𝜃
𝒆𝒆
𝛿𝛿 𝑡𝑡 + 𝜏𝜏𝑟𝑟 = ℎ 𝒆𝒆 𝑡𝑡 + 𝑘𝑘 �̇�𝒆
𝑋𝑋
𝑌𝑌
𝑒𝑒 = 𝑌𝑌𝑅𝑅 − 𝑌𝑌 + 𝐿𝐿 sin𝜃𝜃
𝑚𝑚 (�̈�𝑌 + 𝑢𝑢 �̇�𝜃) = 𝐹𝐹𝑦𝑦1(𝛼𝛼1) + 𝐹𝐹𝑦𝑦2(𝛼𝛼2)𝐼𝐼𝑧𝑧 �̈�𝜃 = 𝐹𝐹𝑦𝑦1(𝛼𝛼1)𝑎𝑎 − 𝐹𝐹𝑦𝑦2(𝛼𝛼2)𝑏𝑏
𝛼𝛼2 = 𝜃𝜃 −�̇�𝑌 − 𝑏𝑏�̇�𝜃
u, 𝜃𝜃 + 𝛿𝛿 − 𝛼𝛼1 =
�̇�𝑌 + 𝑎𝑎 �̇�𝜃𝑢𝑢
𝑦𝑦𝜃𝜃
5 D.O.F. MODEL IN (𝑌𝑌, �̇�𝑌,𝜃𝜃, �̇�𝜃,𝛿𝛿)
�̇�𝑒 = −�̇�𝑌 − 𝐿𝐿 �̇�𝜃 cos𝜃𝜃
+ More human-like response+ Should increase stability+ Add freedom in control design
Bifurcation analysis of an automobilemodel negotiating a curve Rinaldi, Piccardi, Dercole, Gragnani, Miari, Colombo, Della Rossa
DEIB – Dinamica Sistemi Complessi
L [m]
𝑢𝑢 [m/s]
L [m]
𝑢𝑢 [m/s]
𝛿𝛿 𝑡𝑡 + 𝜏𝜏𝑟𝑟 = ℎ 𝒆𝒆 𝑡𝑡 + 𝑘𝑘𝑑𝑑 �̇�𝑒(𝑡𝑡)A more human-like control law take into account also the velocity with which the error changes!
Bifurcation analysis of an automobilemodel negotiating a curve Rinaldi, Piccardi, Dercole, Gragnani, Miari, Colombo, Della Rossa
DEIB – Dinamica Sistemi Complessi
Conclusions and further extensions (master thesis proposal )
• Make the analysis of the vehicle+driver model for different running conditions (steering pad experiment)
• Quantify (in terms of distance from the nearest bifurcation curve) the robustness of a driving style, hence quantifying the ability of a driver
• Use bifurcation analysis in experiments to make a simple and direct comparison between different type of vehicles and different type of drivers
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