Intelligent vehicles and road transportation systems (ITS) Week 2 : - - PDF document

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23-nov.-14 1

Intelligent vehicles and road transportation systems (ITS)

Week 2 : Vehicle dynamics and dynamical models

ME400

Denis Gingras January 2015

D Gingras – ME470 IV course CalPoly Week 2 23-nov.-14 2

 Week 1 : Introduction to intelligent vehicles, context, applications and

motivations

 Week 2 : Vehicle dynamics and vehicle modeling  Week 3: Positioning and navigation systems and sensors (GPS, INS,

• dometer…)

 Week4: Vehicle perception sensing and map building (lidar, radar, sonar,

camera)

 Week 5 : Multi-sensor data fusion techniques  Week 6 : Object detection, recognition and tracking

(Midterm exam Week 1 to 5)

 Week 7: Vehicle control and safe-by-wire basics  Week 7 : ADAS systems and safety applications  Week 8 : The connected vehicles: VANETS, DSRC, V2V, V2I and V2X  Week 9 : Multi-vehicular urban/highway scenarios and collaborative

architectures

 Week 10 : The future: toward autonomous vehicles and automated driving

(Final exam)

Course outline

D Gingras – ME470 IV course CalPoly Week 2

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23-nov.-14 3

 Brainstorming and introduction  Forces acting on road vehicles

 Resistance (Aerodynamic, rolling, grade)  Tractive Effort  Acceleration  Braking and stopping distance

 State space model used in dynamical systems  Vehicular dynamics models

 One wheel (point) model  Two-wheel model  Three-wheel and other full car models

 Driving modes and IMM models

Week 2 outline

D Gingras – ME470 IV course CalPoly Week 2

What is Vehicle Dynamics? Brainstorming

Open questions and introductory discussion

Brainstorming 23-nov.-14 4 D Gingras – ME470 IV course CalPoly Week 2

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Brainstorming

Open questions and introductory discussion

Brainstorming

How many dimensions are we dealing with in describing vehicular dynamics and what are the reference frames involved?

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What are the 3 main branches of classical mechanics used in vehicular dynamics ?

Brainstorming

Open questions and introductory discussion

Brainstorming 23-nov.-14 6 D Gingras – ME470 IV course CalPoly Week 2

4 What can we do with vehicle dynamics? Give some examples

Brainstorming

Open questions and introductory discussion

Brainstorming 23-nov.-14 7 D Gingras – ME470 IV course CalPoly Week 2

Brainstorming

Open questions and introductory discussion

Brainstorming

Define force, torque, work and power in Newtonian mechanics and give their respective units.

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What are the forces acting upon a vehicle ?

D Gingras – ME470 IV course CalPoly Week 2

Brainstorming

Open questions and introductory discussion

Brainstorming Forces 23-nov.-14 10

What are the three main types of motion we find typically in vehicle dynamics and what are the main vehicle subsystems influencing them? Brainstorming

Open questions and introductory discussion

D Gingras – ME470 IV course CalPoly Week 2 Brainstorming

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What are the main driving modes we encounter in vehicular dynamics ?

D Gingras – ME470 IV course CalPoly Week 2

Brainstorming

Open questions and introductory discussion

Brainstorming 23-nov.-14 12

What is resistance and what are the resistive forces?

D Gingras – ME470 IV course CalPoly Week 2

Brainstorming

Open questions and introductory discussion

Brainstorming Forces

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What is a model ?

D Gingras – ME470 IV course CalPoly Week 2

Brainstorming

Open questions and introductory discussion

Brainstorming 23-nov.-14 14

What would be the system elements of a vehicle model in a schema block diagram ?

Brainstorming

Open questions and introductory discussion

Brainstorming D Gingras – ME470 IV course CalPoly Week 2

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What are the main concepts in rectilinear vehicle dynamics ?

Brainstorming

Open questions and introductory discussion

Brainstorming D Gingras – ME470 IV course CalPoly Week 2 23-nov.-14 16

 vehicle’s weight (resulting from gravitational force)  forces which act along the longitudinal axis of the vehicle, e.g.

• motive force,
• aerodynamic drag or rolling friction
• braking force
• steering force,
• centrifugal force when cornering
• Crosswinds
• the chassis (e.g. wind),
• the steering (steering force),
• the engine and transmission (motive force),
• the braking system (braking force).
• braking system (braking force).

Distribution of forces Forces which act laterally on the vehicle The forces are transferred through

Forces acting on a vehicle

D Gingras – ME470 IV course CalPoly Week 2 Forces

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Forces acting upon a vehicle

D Gingras – ME470 IV course CalPoly Week 2

Progression of Friction Coefficient under braking (longitudinally) – µHF is friction coefficient – maximum between 10% and 40% brake slip, (depending on nature of road surface and tires) a): Rising slope of the curve is the “stable zone”

• increase in brake slip

increase in friction b): Falling slope is the “unstable zone”

• increase in brake slip

reduction in friction

Forces 23-nov.-14 18

Forces: Turning forces

Turning Forces act on the entire vehicle  Example μ-split braking  Situation

• wheels on one side of vehicle are on a slippery surface

(e.g. black ice)

• wheels on other side are on a road surface with normal

grip (e.g. asphalt)  Result

• vehicle will slew around vertical axis when the brakes are

applied

• rotation is caused by the yaw moment, which arises due to

the different forces applied to the sides of the vehicle.

D Gingras – ME470 IV course CalPoly Week 2

The different forces involved in the kinematics of a vehicle

Forces

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Forces acting upon a vehicle

D Gingras – ME470 IV course CalPoly Week 2

Coefficient of friction µ

– dependent on the nature of the road surface the condition of the tires the vehicle’s road speed the weather conditions – defines frictional properties of material pairings between tire and road surface and the environmental conditions

Forces

Aerodynamic Resistance Ra

Composed of:

1.

Turbulent air flow around vehicle body (85%)

2.

Friction of air over vehicle body (12%)

3.

Vehicle component resistance, from radiators and air vents (3%)

2

2 V A C R

f D a

 

3

2 V A C P

f D Ra

 

sec 550 1 lb ft hp  

Source: National Research Council of Canada

Forces 23-nov.-14 20 D Gingras – ME470 IV course CalPoly Week 2

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Forces: rolling Resistance Rrl

Composed primarily of

1.

Resistance from tire deformation (90%)

2.

Tire penetration and surface compression ( 4%)

3.

Tire slippage and air circulation around wheel ( 6%)

4.

Wide range of factors affect total rolling resistance

5.

Simplifying approximation:

W f R

rl rl 

        147 1 01 . V frl WV f P

rl rl R



sec 550 1 lb ft hp  

D Gingras – ME470 IV course CalPoly Week 2 23-nov.-14 21 Forces

Forces: grade Resistance Rg

Composed of

 Gravitational force acting on the vehicle

g g

W R  sin 

g g

  tan sin 

g g

W R  tan 

G

g 

 tan

WG Rg 

For small angles,

θg W θg Rg

D Gingras – ME470 IV course CalPoly Week 2 23-nov.-14 22 Forces

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Governing Equation Mass Factor

 (accounts for inertia of vehicle’s rotating parts)

ma R F

m

  

2

0025 . 04 . 1    

m

Forces: Vehicle Acceleration

Forces 23-nov.-14 23 D Gingras – ME470 IV course CalPoly Week 2

Forces: Available tractive effort

The minimum of:

1.

Force generated by the engine, Fe

2.

Maximum value that is a function of the vehicle’s weight distribution and road-tire interaction, Fmax

 

max

, min effort tractive Available F Fe 

D Gingras – ME470 IV course CalPoly Week 2 23-nov.-14 24 Forces

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Forces: tractive Effort Relationships

D Gingras – ME470 IV course CalPoly Week 2 23-nov.-14 25 Forces

Forces: engine-generated tractive effort

Force Power

r M F

d e e

  0 

 

 2 min sec 60 rpm engine 550 lb ft torque sec lb ft 550 hp                 

Fe = Engine generated tractive effort reaching wheels (lb) Me = Engine torque (ft-lb) ε0 = Gear reduction ratio ηd = Driveline efficiency r = Wheel radius (ft)

D Gingras – ME470 IV course CalPoly Week 2 23-nov.-14 26 Forces

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Vehicle Speed vs. Engine Speed

 

1 2   i rn V

e

 

V = velocity (ft/s) r = wheel radius (ft) ne = crankshaft rps i = driveline slippage ε0 = gear reduction ratio

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Typical Torque-Power Curves

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Forces: maximum tractive effort

Front Wheel Drive Vehicle Rear Wheel Drive Vehicle What about 4WD?

 

L h L h f l W F

rl f

     1

max

 

L h L h f l W F

rl r

     1

max 23-nov.-14 29 D Gingras – ME470 IV course CalPoly Week 2 Forces

Diagram

θg

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Vehicle Acceleration

Governing Equation Mass Factor

 (accounts for inertia of vehicle’s rotating parts)

ma R F

m

  

2

0025 . 04 . 1    

m

D Gingras – ME470 IV course CalPoly Week 2 23-nov.-14 31 Forces

Braking Force

Front axle Rear axle

   

L f h l W F

rl r bf

    

max

 

 

L f h l W F

rl f br

    

max

D Gingras – ME470 IV course CalPoly Week 2 23-nov.-14 32 Forces

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Braking Force

Ratio Efficiency

   

rear front f h l f h l BFR

rl f rl r

         

max

g

b 

D Gingras – ME470 IV course CalPoly Week 2 23-nov.-14 33 Forces

Vehicle/Tire Model Interaction

VEHICLE MODEL Wheel centre - Position, Orientation and Velocities  Mathematical Solution at Integration Time Steps  TIRE MODEL Fx - longitudinal tractive or braking force Fy - lateral cornering force Fz - vertical normal force Mz - aligning moment Mx - overturning moment My - rolling resistance moment

Tire Model Fy Fx Fz Mz Tire Model Fy Fx Fz Mz

Forces: tire

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Forces: tire

D Gingras – ME470 IV course CalPoly Week 2

Tire forces

– A vehicle can only be made to move

• r change its direction in a specific way

by forces acting through the tires. – Friction (= adhesion)  between tire and road surface determines the wheel’s ability to transmit force. – FB = µ ∙ FN – ABS and TCS utilize the available adhesion to its maximum potential.

Forces 23-nov.-14 36

State space model

State-space model

Dynamical systems are usually expressed in recursive form. The “state space model ” is a widely used linear recursive model for describing dynamical systems. It is expressed as follow:

( 1) ( ) ( ) x k Ax k w k   

( ) ( ) ( ) ( ) z k H k x k k   

where: = state variables, (n x 1) column vector = measurements, (m x 1) column vector = state transition matrxi, (n x n) matrix = states to measurements matrix, (m x n) matrix = state transition noise, (n x 1) column vector = measurement noise, (n x 1) column vector

( ) ( ) ( ) ( ) x k z k A H w k k 

D Gingras – ME470 IV course CalPoly Week 2

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State space model

In vehicular dynamics, the states are usually physical quantities of interest , such as (linear and angular) position, velocity, acceleration, etc. As a first approximation, all the noises are often assumed to be white, which is not always the case. is

• ften referred to the model noise, which is inducted to the system and affects the

state variables. is the noise coming from the sensors. ( ) w k ( ) k  From the state space model equations, we can obser that the next state value depends only on the present state value and the current measurement. Therefore the model assumes that the system is Markovian, that is, all the system history, previous to the current state value, does not influence the next state value. Both matrices A and H are with all elements being constant. The matrix A describes how the system changes with time. It contains the equations of motion of the vehicle, expressed as a 1st order difference (discrete) or differential (continuous)

• equations. The matrix H describes the relationship between the measurements and

the state variables. It contains the sensor information and defines how each state variable is mapped into the measurements.

State-space model D Gingras – ME470 IV course CalPoly Week 2 23-nov.-14 38

State space model

Because both noises are assumed to be white, they are zero-mean uncorrelated stochastic processes. Therefore, the only thing we need to describe the noises are their covariance matrix. The two covariance matrices are diagonal: = model noise covariance matrix of , (n x n) diagonal matrix = measurement noise covariance matrix of , (m x m) diagonal matrix

( ) w k

Q R

( ) k 

The main diagonal elements correspond to the variance of the noise random variables (since the noise are assumed iid). To use the state space model adequately, we need to design both matrices A and H , and we need to estimate as accurately as possible the noise covariance matrices. The model is used extensively in multi-sensor data fusion (Week 5) and control (Week 7). The most common approach to estimate the state variables using this model is the Kalman Filter.

State-space model D Gingras – ME470 IV course CalPoly Week 2

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Vehicle motion models

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Vehicle motion models can be divided into two subcategories: the uniform motion model and the maneuvering model. Furthermore, vehicle models can vary according to the level of physical details taken into account. For example, in turning maneuvers, using a two-wheel model, we have the: Kinematic vehicle models: the kinematic vehicle model is based on the assumptions of zero lateral velocity and no wheel slip. This model is suitable for low-speed and low-slip driving conditions. However, in the high-speed and large-slip driving region, the assumptions of the kinematic vehicle model break down, resulting in poor estimation performance. Dynamic vehicle models: in contrast to the kinematic vehicle model, the dynamic vehicle model considers lateral tire force and vehicle

• moments. Hence, it can estimate the vehicle lateral velocity and

wheel slip, and can be applied to high-speed and large-slip driving conditions.

Modeling dynamics D Gingras – ME470 IV course CalPoly Week 2 23-nov.-14 40

In the road plane, vehicular motion is described by a combination of two basic movements: rectilinear and angular. For a rectilinear movement, we define :  Longitudinal movement of the center of gravity  Lateral movement  Vertical movement. For an angular movement, we have:

• Heading (yaw) angle
• Roll angle
• Pitch angle

For a 4-wheel vehicle, we have 6 degrees of freedom (degrees of movement in 3D space).

Vehicle dynamics

D Gingras – ME470 IV course CalPoly Week 2 Modeling dynamics

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• Longitudinal dynamics relates to longitudinal rectilinear movement. The

vehicle is considered as a rigid body with mass m moving along the road. Neglecting the resistive aerodynamic forces, if the vehicle is moving at speed v, and applying a traction force fd , taking into account road friction, we have,

d

mx x f     

m

D Gingras – ME470 IV course CalPoly Week 2

Vehicle dynamics

v  is the road friction coefficient applied to the 2 wheels in this example.

Assuming zero initial conditions, with the vehicle moving at constant velocity with a fixed traction force, the equation becomes, In this case, the speed is inversely proportionnal to  .

d d

f x f x       

Modeling dynamics 23-nov.-14 42

1D case: With initial position x0 at initial time t0, the vehicle position x at

a given time t is given by:

Position Speed (v) Acceleration (a)

( ) ( ) ( ) ( ) ( ) ( ) x t x t t t x t t t x t        

( ) dx t v x dt   

2 2

( ) d x t a x dt   

( ) x t

D Gingras – ME470 IV course CalPoly Week 2

Vehicle dynamics

Lateral dynamics deals with angular movement of the vehicle. It is mainly considered for questions of stability and vehicle control in presence of high lateral speed and acceleration. The study of lateral dynamics is usually performed using the two-wheel (bicycle) model.

Modeling dynamics

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Using general kinematics motion equations to describe the longitudinal dynamics of the vehicles, which, after discretization leads to, where n is the simulation step, T the simulation time step (samling), xi the longitudinal position of vehicle i, vi the speed of vehicle i, and ai the acceleration of vehicle i. The acceleration used is calculated according to the current driving situation and is limited to a maximum acceleration and deceleration comfortable to human.

1D longitudinal vehicle dynamics model

2

( 1) ( ) ( ) 0.5 ( ) ( 1) ( ) ( )

i i i i i i i

x n x n v n T a n T v n v n a n T       

Modeling dynamics D Gingras – ME470 IV course CalPoly Week 2 23-nov.-14 44

Kinematic model for rectilinear motion

In rectilinear motion, the vehicle dynamics can be modeled as a piecewise constant Wiener process acceleration (DWPA) model. The corresponding state equation (1D) are, With And for the third-order state equation,

 

( ) ( ) ( ) ( )

T

X n x n x n x n   

Model with acceleration increment process noise

( ) ( 1) ( 1) X n FX n n      

2

1 1 2 1 1 T T F T               

2

1 2 1 T T                

Where (n) is the so-called acceleration increment during the nth sampling period and it is assumed to be a zero-mean white process with variance σ2.

Modeling dynamics D Gingras – ME470 IV course CalPoly Week 2

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Kinematic model for rectilinear motion

Assuming that accelerations in the steady state are quite small (abrupt motions like a sudden stop or a collision are not considered), linear accelerations or decelerations can be reasonably well covered by process noises with the constant velocity model. That is, the constant velocity model plus a zero- mean 2D noise vector,, with covariance matrix, , representing the magnitude of acceleration can handle uniform motions on the road. In discrete- time, for the constant velocity model with noise and sampling time T, the kinematic equations (state vector) become,

2 2

1 ( ) 1 2 ( ) 1 ( ) ( 1) ( 1) 1 ( ) 1 2 ( ) 1 x n T T x n T X n X n n y n T T y n T                                                

Small acceleration case

( ) ( 1) ( 1) X n FX n n      

2 T

Q



   

Modeling dynamics D Gingras – ME470 IV course CalPoly Week 2 23-nov.-14 46

The major assumption used in the development of this two-wheel kinematic model is that the velocity vectors at points A and B are in the direction of the orientation

• f the front and rear wheels respectively. This is reasonable at low speed.

Source: R Rajamani, “Vehicle Dynamics and Control”, Springer, 2nd Ed, 2012

Kinematics of lateral vehicle motion

Modeling dynamics D Gingras – ME470 IV course CalPoly Week 2

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The turn of most vehicles usually follows a pattern known as coordinated turn (CT), characterized by a (nearly) constant turn rate and a (nearly) constant speed. Assuming the turn is taking place in an horizontal plane (2D), a CT is a turn with a constant yaw rate along a road of constant radius of curvature. However, the curvatures of actual roads are not constant. Hence, a fairly small noise is added to a constant-speed turn model for the purpose of capturing the variation of the road

• curvature. The noise in this model represents the modeling error, such as the

presence of angular acceleration and non-constant radius of curvature. For a vehicle turning with a constant angular rate and moving with constant speed (the magnitude

• f the velocity vector is constant), the kinematic equations for the longitudinal and

tangential acceleration in the (x,y) plane are,

Kinematic model for coordinated turn

( ) ( ), ( ) ( ) x n y n y n x n         

Where  represents the yaw rate (also called the turn rate or the angular rate). Note: ω > 0 implies a counter-clockwise turn). The tangential component of the acceleration is equal to the rate of change of the speed, that is, and the normal component is defined as the square of the speed in the tangential direction divided by the radius of the curvature of the path, that is,

   

( ) ( ) ( 1) / ( ) ( 1) / y n y n y n T x n x n T          

2 2

( ) ( ) / ( ) ( ), ( ) ( ) x n y n x n x n with y n x n          

Modeling dynamics D Gingras – ME470 IV course CalPoly Week 2 23-nov.-14 48

Kinematic model for coordinated turn

Note that the noise vector here is 3D, the first two for modeling the longitudinal acceleration, whereas the 3rd one for the angular acceleration.

2 2

1 ( ) 1 sin ( ) / ( ) (1 cos ( ) ) / ( ) 2 ( ) cos ( ) sin ( ) ( ) ( 1) ( ) (1 cos ( ) ) / ( ) 1 sin ( ) / ( ) 1 2 ( ) sin ( ) cos ( ) 1 T x n n T n n T n x n n T n T T X n X n y n n T n n T n T y n n T n T T T                                                            ( 1) n                  

 

( ) ( ) ( ) ( ) ( ) ( )

T

X n x n x n y n y n n    

Assuming the angular speed to be piecewise constant during the turn, the state equations become,

Modeling dynamics D Gingras – ME470 IV course CalPoly Week 2

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Vehicle path angle in lateral kinematics

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Source: ACAS Program, final report, executive summary, 1998.

D Gingras – ME470 IV course CalPoly Week 2 Modeling dynamics 23-nov.-14 50

Two-wheel model

The two-wheel (bicycle) model is used mainly to simplify the study of lateral dynamics of a vehicle. The model is based on two degrees of freedom: lateral velocity and angular velocity following the vertical axis. Steering angle δ and engine traction force are considered as inputs to the

• model. By applying the 2nd law of Newton following the y (lateral) axis, we

have : Where lateral acceleration, , is the sum of the acceleration due to lateral movement following the y axis and the centripet acceleration .

y yf yr

mv F F   

y

v 

x

V  

y 

y

V  

yf yr y x

F F V V a b  Force applied to frontwheel Force applied to rear wheel Lateral speed Longitudinal speed Distance between center of gravity and front axle Distance between center of gravity and rear axle Steerin   g angle applied to front wheel Angular speed of vehicle heading D Gingras – ME470 IV course CalPoly Week 2 Two-wheel model

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The lateral forces are:

( )

x yf yr

m y V F F      

y f y r

I a F b F    

2 ( * )

yf f f

F k  

y f x

v a V       

k

y r x

v b V      

2 ( * )

y r r r

F k  

D Gingras – ME470 IV course CalPoly Week 2

Two-wheel model

The equation then becomes, By applying the sum of inertial moments, following the axis perpendicular to the model, and going through the center of gravity, we obtain, where I is the rotation inertial of the de l’axe. and = rigidity factor (stifness). and are the rear and front wheels slip angles

Two-wheel model 23-nov.-14 52

2 2

( 2 ) (2 ) 2 2 2

f r f r f x y y x f r f f r x x

ak bk k k k v v v m v m vx m ak bk ak a k b k Iv I Iv                                                         

D Gingras – ME470 IV course CalPoly Week 2

From these equations, the linear bicycle model can be put in a state space model form. Information provided by this dynamical model are used for lane tracking and following as well as for applications in stability control.

Two-wheel model

Two-wheel model

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Two-wheel model

Two-wheel model

Typical use of bicycle model: linear state equation relating yaw and lateral speed with steering angle, lateral speed after a step steering

D Gingras – ME470 IV course CalPoly Week 2 23-nov.-14 54

Two-wheel model: kinematic model.

An assumption of this model is that there is no slip between the wheels and the ground. Therefore, the steering angle is directly related to the yaw rate.

The kinematic vehicle model is derived from the two degrees of freedom of the fundamental bicycle model In the bicycle model, the left and right front/rear wheels of a vehicle are represented by a single wheel.The front steering angle is described by δ . The center of gravity of the vehicle is at point G . The distances from G to the front and rear wheels are represented by lf and lr , respectively. The (X ,Y) coordinates represent the location of G in the global frame, and ψ describes the orientation of the vehicle. The VG represents the velocity at the G and it makes the slip angle β .

Source: J Kichun et al., “Integration of Multiple Vehicle Models with an IMM Filter for Vehicle Localization”, IEEE IV Symposium, 2010.

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Two-wheel model: kinematic model.

Under low-speed and small-slip driving conditions (< 5 to 10 km/h) the kinematic vehicle model is better because it is simpler and it neglects the noise process of the lateral velocity of the vehicle. Therefore a positioning estimator filter based on this model at low speed is more robust and provide a better estimate of the position.

The kinematic equations become

1

cos( ) tan( ) sin( ) , tan ( ) cos( )tan( ) / ( )

G r G f r G f r

X V l Y V where l l Z V l l        



                                    

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Two-wheel model: dynamic model.

This model takes into account slipping between the wheels and the ground. It assumes that the lateral force that acts on a tire is proportional to the tire slip angle.

The tire slip angle can be defined as the angle of the wheel velocity vector relative to the longitudinal wheel axis. The tire slip angle of the front and rear wheels are represented by αf and αr respectively. m and Iz describe the vehicle mass and yaw moment of inertia while Vx and Vy represent the vehicle longitudinal and lateral velocities respectively. The equations describes the lateral dynamics of the bicycle model. It is derived by summing the forces and moments about G, the center of gravity. The front steering angle is described by δ.

Source: R. Rajamani, Vehicle dynamics and control: Springer, 2006

( )

y yf rf y x z f yf r yr z

F F F ma m V V r M l F l F I         

 

 

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Two-wheel model: dynamic model.

Under high-speed and large-slip driving conditions, the dynamic vehicle model is more appropriate because it considers lateral velocity and the slip of a vehicle. Cf and Cr represent the front and rear tire cornering stiffness respectively.

The dynamic equations become

2 2

cos( ) sin( ) cos( ) sin( ) 2 2 2 2 2 2 2 2 2 2

x y y x f f r r f f r r f f y z x z x z y f f r r f r f x y x x

V V V V X C l C l C l C l C l Y V I V I V I V C l C l C C C V V mV mV m                                                                                                  ,      

2 2 , 2 2

f r yf f f f yr r r r x x

l l F C C F C C V V                            

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Holonomic and nonholonomic systems

They are mechanical systems in which all links are geometrical (holonomic)— that is, restricting the position (or displacement during motion) of points and bodies in the system but not affecting the velocities of these points and bodies. For example, a train on rails. Consider a two-dimensional space; the degrees of freedom are the x axis, y axis, and rotation about the origin. In this space, a mobile base with three

• mnidirectional wheels in a triangular configuration would be considered

holonomic, but a tricycle would not (due to the Parallel parking, see next slides). A non-holonomic system is a system whose state depends on the path taken to achieve it, such as a car. It is described by a set of parameters subject to differential constraints, such that when the system evolves along a path in its parameter space but finally returns to the original set of values at the start

• f the path, the system itself may not have returned to its original state. The

division of mechanical systems into holonomic and non-holonomic is most significant, since a number of equations that make possible the comparatively simple solution of mechanical problems are applicable only to holonomic systems.

Source: Wikipedia

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The parallel parking problem: is a motion planning problem in control theory and vehicle dynamics to determine the path a car must take in order to parallel park into a parking space. The front wheels of a car are permitted to turn, but the rear wheels must stay fixed. When a car is initially adjacent to a parking space, to move into the space it would need to move in a direction perpendicular to the allowed path of motion of the rear wheels. The admissible motions of the car in its configuration space are an example of a non- holonomic system.

Holonomic and nonholonomic systems

Source: Wikipedia

holonomic systems

Source: D Gruyer et al., LIVIC, IFSTTAR

D Gingras – ME470 IV course CalPoly Week 2

Stopping distance = Braking distance + Perception distance+ Reaction distance

For truck The Braking-distance is a function of several parameters:

• The linear and angular velocities
• The road-vehicle interface friction
• The slope of the road
• The weight of the vehicle
• Braking system capabilities of the vehicle
• Aerodynamic resistance
• Coefficient of friction
• 2

1

• COMPUTATION OF THE STOPPING DISTANCE

1) Braking distance

Stopping Distance

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perception time: is how long the driver takes to see the hazard, and the brain realize it is a hazard requiring an immediate reaction Reaction time: is how long the body takes to move the foot from accelerator to brake pedal.

• 2)Perception distance and Reaction Distance:

Stopping Distance

Stopping distance D Gingras – ME470 IV course CalPoly Week 2 23-nov.-14 62

Manual braking of a vehicle

 Recognition time ≈ 0.1s  Pre-braking time ≈ 1.0s

• reaction time (focused) ≈ 0.2-‐0.3s
• reaction time (unfocused) ≈ 0.5-‐0.7s
• conversion time (foot) ≈ 0.2s
• response time ≈ 0.2s
• pressure buildup/2 ≈ 0.1-‐0.2s

 Braking time

• depends on braking power and friction

Stopping time = pre-braking time + braking time

D Gingras – ME470 IV course CalPoly Week 2 Stopping distance

Stopping Distance

32 Braking

 Maximum braking force occurs when the

tires are at a point of impending slide.

 Function of roadway condition  Function of tire characteristics

 Maximum vehicle braking force (Fb max) is

 coefficient of road adhesion () multiplied by the

vehicle weights normal to the roadway surface

Stopping Distance

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Braking Force

 Maximum attainable vehicle deceleration is g  Maximum obtained when force distributed as per weight

distribution

 Brake force ratio is this ratio that achieves maximum braking

forces

Stopping Distance

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 Theoretical

 Assumes effect of speed on coefficient of rolling resistance is

constant and calculated for average of initial and ending speed

 Ignores air resistance  Minimum stopping distance given braking efficiency

 For population of vehicles, what do you assume about rolling resistance, coefficient of adhesion, and braking efficiency?

 

2 2 1 2

2 ( sin )

b b rl g

V V S g f        

Stopping Distance

Stopping distance 23-nov.-14 65 D Gingras – ME470 IV course CalPoly Week 2

Stopping Distance

Theoretical

 ignoring air resistance

Practical Perception Total

           G g a g V V d 2

2 2 2 1 p p

t V d

1



p s

d d d   a V V d 2

2 2 2 1 



For grade = 0

D Gingras – ME470 IV course CalPoly Week 2 23-nov.-14 66 Stopping distance

 

2 2 1 2

2 ( sin )

b b rl g

V V S g f        

34

Stopping Sight Distance (SSD)

Worst-case conditions

 Poor driver skills  Low braking efficiency  Wet pavement

Perception-reaction time = 2.5 seconds Equation

r

t V G g a g V SSD

1 2 1

2           

23-nov.-14 67 D Gingras – ME470 IV course CalPoly Week 2 Stopping distance

Stopping Sight Distance (SSD)

Source: ASSHTO A Policy on Geometric Design of Highways and Streets, 2001

Note: this table assumes level grade (G = 0)

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 

   

 

a V V V V G g a g V V d

2 2 2 2 2 1 2 2 2 1

075 . 1 2 . 11 075 . 1 2 . 11 1 2 47 . 1 2 . 32 2 . 11 2 . 32 2 47 . 1 2             

• 1. Acceleration due to gravity, g = 32.2 ft/sec2
• 2. There are 1.47 ft/sec per mph
• 3. Assume G = 0 (flat grade)

p p p

Vt t V d 47 . 1 47 . 1

1

   

V = V1 in mph a = Deceleration, 11.2 ft/s2 in US customary units tp = Conservative perception / reaction time = 2.5 seconds

p s

Vt a V d 47 . 1 075 . 1

2

 

Stopping Sight Distance (SSD)

Example

Stopping distance 23-nov.-14 69 D Gingras – ME470 IV course CalPoly Week 2

Tires

The tire is the connecting link between the vehicle and the road. It is at that point that the safe handling of a vehicle is ultimately decided. The tire transmits motive, braking and lateral forces within a physical environment whose parameters define the limits of the dynamic loads to which the vehicle is subjected.

Increase in braking distance on wet road surface as a function

• f tire tread depth at

100 km/h.

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Stopping Sight Distance (SSD)

 Worst-case conditions

 Poor driver skills  Low braking efficiency  Wet pavement

 Perception-reaction time = 2.5 seconds  Equation

r

t V G g a g V SSD

1 2 1

2           

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SSD – Quick and Dirty

 

   

 

a V V V V G g a g V V d

2 2 2 2 2 1 2 2 2 1

075 . 1 2 . 11 075 . 1 2 . 11 1 2 47 . 1 2 . 32 2 . 11 2 . 32 2 47 . 1 2             

• 1. Acceleration due to gravity, g = 32.2 ft/sec2
• 2. There are 1.47 ft/sec per mph
• 3. Assume G = 0 (flat grade)

p p p

Vt t V d 47 . 1 47 . 1

1

    V = V1 in mph a = deceleration, 11.2 ft/s2 in US customary units tp = Conservative perception / reaction time = 2.5 seconds p s

Vt a V d 47 . 1 075 . 1

2

 

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Full car model

Modeling dynamics

They have a much higher level degree

• f freedom (typically 12 to 14) than

bicycle models (3). They are typically used for detailed modeling of sub- systems, for chassis control or for electronic stability systems (ESC).

The Ackerman steering model: this is used with the 4-wheel full car model in detailed study of vehicular stability and control.

Source: A Broggi et al., “Handbook of robotics”, Ch 51, Intelligent vehicles, Springer, 2008

Turning dynamics models

Modeling dynamics

Above: A 2-wheel model for vehicle turning with no roll.

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Differential steer from a trapezoidal tie-rod arrangement (4-wheel model)

Source: R Rajamani, “Vehicle Dynamics and Control”, Springer, 2nd Ed, 2012

Turning dynamics models

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Typical driving patterns of vehicles

Straight line and curve Cut-in and cut-out U-turn Interchange

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These modes can be used to construct a dynamic model based on a markov chain. The markov chain models are used in data fusion (for example the IMM(Interactif Multiple Model) to estimate the vehicle dynamics along a given path. CA TR TL BW

Stop

CV

D Gingras – ME470 IV course CalPoly Week 2

 Stopped (vehicle at rest)  Constant speed rectilinear (CV)  Constant angular speed turning right (TR)  Constant angular speed turning left (TL)  Constant acceleration rectilinear (CA)  Reverse constant speed (BW)

Multiple mode dynamic model

Suppose we define the following 6 modes

Modeling dynamics Multiple-mode models

Conclusion

 A vehicle is mostly seen as a rigid body (no trailer).  In intelligent vehicle applications, we focus mainly on longitudinal

and lateral motion, thus ignoring vertical motion usually.

 The state space model is mostly used for modeling vehicular

dynamics

 Stopping distance computation is an essential capability for

intelligent vehicle safety related applications.

 The level of details in modeling the vehicle dynamic depends of

the context and the application. For example, in traffic analysis, individual vehicles are simply modeled as individual particles, whereas in designing stability control systems, a highly refined dynamical model is used.

 Defining driving modes allows to simplify the problem and use

different state space models for different driving modes, which circumvents the problem of dealing with highly nonlinear models.

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References

 R. Rajamani, Vehicle dynamics and control: Springer, 2006.  Li Li & Dr. Fei-Yue Wang (Ed.), Advanced Motion Control and Sensing for

Intelligent Vehicles, Springer 2007.

 B. Siciliano et al. (Ed.), Handbook of robotics, Chapter 51, “Intelligent

vehicles”, Springer, pp. 1175-1198, 2008

 Blundell M, The Multibody system approach to vehicle dynamics, Elsevier,

2004

 Ellis, John Ronaine Vehicle handling dynamics / by John R. Ellis 1994  Gillespie, Thomas D. Fundamentals of vehicle dynamics / Thomas D.

Gillespie

 Vehicle System Dynamics: International Journal of Vehicle Mechanics and

Mobility, Taylor and Francis.

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QUESTIONS?

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