LuGre_Tire_Model

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ARM Internal PresentationNovember, 2006

Dynamic Tire-Road Friction Model for the Dynamic Tire-Road Friction Model for the Simulation of Vehicle Handling Simulation of Vehicle Handling

PerformancePerformance

Ragnar LedesmaRagnar LedesmaCVS Advanced Engineering

ArvinMeritor, Inc.

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Problem Statement

• Pacejka tire model requires extensive tire testing for every tire being considered

• Pacejka coefficients do not have physical significance – they are essentially curve fitting parameters

• Pacejka coefficients are not readily available – tire manufacturers sometimes provide raw data showing lateral forces and aligning moments versus slip angles

• We need a tire model based on first principles, i.e., model the mechanics of friction between the tire and the road

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Objectives

• Develop an analytical tire model for vehicle handling simulation, subject to the following requirements:• Minimum number of states (internal variables) for fast run-times

• Minimum number of model parameters

• Model parameters should have physical meaning

• Accurate in both transient and steady-state conditions

• Capable for combined braking and turning maneuvers

• Captures hysteresis effects and tire relaxation lags

• Tire-road friction forces can be scaled with normal force

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LuGre Tire-Road Friction Model

• Dynamic friction model for determining the tire-road friction forces along the fore-aft and lateral directions

• Model inputs include: • Tire normal force

• Translational velocity of wheel center

• Angular velocity of wheel

• Tire slip angle

• Model outputs include:• Tire longitudinal force

• Tire lateral force

• Tire aligning moment

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LuGre Tire Model Parameters

• Tire-road friction parameters• Static friction coefficient

∀ µsx – along x-axis (longitudinal direction)

∀ µsy – along y-axis (lateral direction)

• Kinetic friction coefficient

∀ µkx – along x-axis (longitudinal direction)

∀ µky – along y-axis (lateral direction)

• Stribeck velocity, vs

• Stribeck exponent, γ

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• Brush model parameters (model for carcass and thread deformation) • Stiffness w.r.t. elastic deformation of bristles

∀ σ0x – along x-axis (longitudinal direction)

∀ σ0y – along y-axis (lateral direction)

• Viscous damping rate w.r.t. elastic deformation of bristles



• Equivalent viscous damping rate representing the Coulomb friction due to sliding between the tire and the road



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• Tire geometry parameters • Static loaded radius, R

• Contact patch length, L – function of static loaded radius and tire radial deformation

• Tire-road contact pressure distribution, fn(ξ)

• Uniform

• Trapezoidal

• Parabolic

• Trigonometric

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LuGre Tire Model

• LuGre tire model can be expressed in 3 forms:• Distributed model (infinite number of states)

• Discrete model (large, finite number of states)

• Average lumped model (minimum number of states)

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Distributed LuGre Tire Model

• Partial differential equations for bristle deformation

• Expressions for the instantaneous friction coefficient

• Expressions for the longitudinal and lateral friction forces

• Expression for the tire aligning moment

ξξωξ

µσλξ

∂∂

−−=∂

∂ ),(),(

)(),(2

0 tzRtz

vv

t

tz ii

ki

irri

iyxi ,=

riii

iiii vt

tztzt *

),(*),(*),( 210 σξσξσξµ +

∂∂

+= yxi ,=

ξξξµ dfttF n

L

ii *)(*),()(0∫= yxi ,=

ξξξξµ dLfttM n

L

yz *)2/(*)(*),()(0

−= ∫

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Discrete LuGre Tire Model





)(1)(

),1(,,20

, ijijijki

irriij zz

L

NRz

vvz −−−−−= ω

µσλ

,,...,2 Nj = ,0,1 =iz yxi ,=

riiijiijiij vzz *** 2,1,0, σσσµ ++= ,,...,2 Nj = yxi ,=

∑= −

=N

jjniji N

LfF

2,, )

1(*µ yxi ,=

∑= −

−−=N

jjnijz N

LNLjLfM

2,, )

1(*))1/(*2/(**µ

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Average Lumped LuGre Tire Model





iiiki

irrii ztRz

vvz *)(*

)(2

0 κωµ

σλ −−=

LzRztRzv

vLF

Gz ytit

ky

yrry

nt /**)(*

)(2

0 ωυωµ

σλ+−−=

yxi ,=

riiiiiii vzz *** 210 σσσµ ++=

)***()( 210 riiiiiini vzzFtF σσσ ++= yxi ,=

)}2

1(*)

2

1(*)

2

1(*{)( 210 LF

GvzzzzLFtM

nryytyytyynz −+−+−= σσσ

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Example: Lu-Gre Tire Model Parameters

• Tire-road dynamic friction model parametersLongitudinal direction (x-axis):

σx0 555 (1/m) – stiffness w.r.t. bristle elastic deformation σx1 0.033 (sec/m) – damping w.r.t. bristle deformation rate σx2 0.001 (sec/m) – equiv. viscous damping w.r.t. sliding velocity µsx 1.35 – static friction coefficient µkx 0.75 – dynamic friction coefficient

Lateral direction (y-axis): σy0 470 (1/m) – stiffness w.r.t. bristle elastic deformation σy1 0.033 (sec/m) – damping w.r.t. bristle deformation rate σy2 0.001 (sec/m) – equiv. viscous damping w.r.t. sliding velocity µsy 1.40 – static friction coefficient µky 0.75 – dynamic friction coefficient

Other parameters vs 3.96 (m/sec) – Stribeck velocity γ 1.0 – Stribeck exponent L 0.15 (m) – contact patch length R 0.50 (m) – static loaded radius

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Example: Longitudinal Forces

• Fx versus slip ratio curves (constant slip angle)

Fn=2000 N

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Example: Friction Ellipse (combined slip)

• Fx versus Fy curves (constant slip angle)

Fn=2000 N

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Example: Lateral Forces

• Fy versus slip angle (constant slip ratio)

Fn=22,240 N

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Example: Aligning Moment

• Mz versus slip angle (constant slip ratio)

Fn=22,240 N

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Next Steps

• Implement the tire model in TruckSim

• Compare results with built-in TruckSim tire model

• Check if tire relaxation effects are captured

• Implement tire model in ADAMS• Average lumped model

• Discrete LuGre tire model

LuGre_Tire_Model

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