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

7

- 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

15

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

20

Optimal control – Single track

21

𝐽′

Optimal control – Single track

22

Single-track (bicycle) model

23

•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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37

Thank you!

Questions?