Adithya Arikere - sveafordon.com · For our purposes: Mathematical model “mathematical (symbolic)...

Adithya Arikere

Financed by FFI

•Find opportunities for novel or

enhanced active safety

functionality enabled by electric

drives • Study dynamics of accident scenarios

• Design and implement controllers for

interventions

• Validate controllers in simulations

• Validate controllers in experiments

Project objective

2

•For our purposes: Mathematical model

“mathematical (symbolic) representation of a concept, phenomenon,

relationship, structure, system, or an aspect of the real world”

•Can be used to explain, control or predict system behaviour based on inputs

or past observations

What is a model?

3

No!

Do we need models?

4

•Versatility

•Online evaluation/computation/adaptation

•Extensibility

•Interactive

•Analysis and understanding

•Build and test hypothesis

•Generate theorems/proofs

•Generate simple solutions

•Knowledge is transferable

•...

Do we want models?

5

•Active safety

•Interventions just a second or two before a crash to

prevent or mitigate the same

•On- or near-limit interventions

•Highly non-linear dynamics

•Cost of failure can be high

Model requirements

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“All models are wrong, but some are useful”

Model requirements

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- George Box (Statistician)

•Different ways to classify:

•Point mass, single-track, two-track

•Analytical/Numerical

•Monolithic/Modular

•Tyre model type

•Degrees of freedom

•...

Classification of models

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• Reference frame: Global

• Friction circle representation: Quadratic constraint (Cartesian coordinates)

Point mass (particle) model

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𝑋

𝑌 𝑌

𝑋

𝑋

𝑌 𝐹𝑌

𝐹𝑋

• Reference frame: Global

• Friction circle representation: Linear constraint (Polar coordinates)

Point mass (particle) model

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𝑋

𝑌 𝑌

𝑋

𝑋

𝑌

𝐹

𝜙

Point mass (particle) model

11

• Reference frame: Local

• Friction circle representation: Quadratic constraint (Cartesian coordinates)

𝑋

𝑌 𝑢

𝑋

𝑌

𝐹𝑥

𝐹𝑦

𝜈

Point mass (particle) model

12

• Reference frame: Local

• Friction circle representation: Linear constraint (Polar coordinates)

𝑋

𝑌 𝐹

𝜙

𝑋

𝑌 𝑢

𝜈

Intersection accidents

Opponent Host

Left Turn Across Path – Opposite

Direction (LTAP/OD) 13

13

Intersection accidents

Opponent Host

𝒅

Distance margin

𝒗𝒃

𝒗𝟎

𝒀𝒃

Driver assist interventions only

Steering always performed by driver 14

14

Optimal control solution

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Global reference frame Local reference frame

States (𝒙)

Hamiltonian

(𝐻)

Control

inputs (𝒖)

Costate

rate (𝜆 )

Costates

(𝜆) ? ? ? ? ?

⋮ ⋮ ⋮

Optimum fo-

rce angle (𝜙) 𝑵𝒐𝒕 𝒔𝒐𝒍𝒗𝒂𝒃𝒍𝒆

Single-track (bicycle) model

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Equations of motion: Slip angles:

Linear single-track (bicycle) model

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•Simplification: • Longitudinal velocity constant

• Linear tyres

Linear single-track (bicycle) model

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•Simplification: • Longitudinal velocity constant

• Linear tyres

• Torque vectoring

Linear single-track (bicycle) model

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• FWS = Front Wheel Steer

• DYC = Direct Yaw Control (no steering, torque vectoring only)

• YRC = Front Wheel Steering (steady state) + Torque Vectoring (transient)

Optimal control – Single track

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Optimal control – Single track

21

𝐽′

Optimal control – Single track

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Single-track (bicycle) model

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•Common variations: • Constant longitudinal velocity

• Torque vectoring

• Steady state

• Longitudinal slip

• Longitudinal load transfer (dynamic or steady state)

• Lateral load transfer (tyre stiffness and capacity adaptation based on lateral acceleration)

• ....

Single-track (bicycle) model

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•Use cases: •Understeer/oversteer characteristics

• Steady state or transient yaw or lateral acceleration response

• Stability and controllability analysis

•Reference generation • Commonly used as a yaw rate reference generator for ESC or other such

controllers

• Simplest model that captures yaw degree of freedom

• Validation of point mass results

• Analytical studies

Two-track model

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Equations of motion:

Two-track model

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Slip angles:

Two-track model

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• Lateral transient load transfer: • Adds roll degree of freedom

• Steady state longitudinal load transfer

• Total wheel load

Optimal control – Two track

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Optimum global force angle from numerical optimal control

Two-track model

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•Common variations: • Wheel degrees of freedom, longitudinal slip (+4 DoF)

• Pitch and roll degrees of freedom (+2 DoF)

• Suspension modelling

• Steering models

• Rear-wheel steering or individual wheel steering

• Powertrain models

• Front, rear, all-wheel drive, torque vectoring, etc

• Steady state

• ....

Two-track model

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•Use cases: • Preliminary testing of vehicle subsystems (engine, steering,

suspension, etc)

• Preliminary testing of active safety (or other control) functions

• Yaw stability analysis or verification of stability characteristics

• Simplest model that contains all major features influencing yaw stability

• Validation of point mass results

Tyre models

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Magic formula tyre model

Tyre models

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Magic formula tyre model

Tyre models

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Magic formula tyre model

Tyre models

35

tanh tyre model

Tyre models

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Linear tyre model

•Reference generation / preliminary analysis almost always with point mass model

•Single-track / two-track model useful for verification and analysis

•Choice of tyre-model may be important

•High-fidelity models almost always for validation, never for reference generation

•Experiments ultimate validation tool

Summary

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Thank you!

Questions?