prevention of tripped rollovers

8/16/2019 prevention of tripped rollovers

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Prediction and Prevention of Tripped Rollovers

Final Report

Prep a red b y :

Gridsada Phanomchoeng

Rajesh Rajamani

Department of Mechanical EngineeringUniversity of Minnesota

CTS 12-3 3


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Technical Report Documentation Page 1. Report No. 2. 3. Recipients Accession No. CTS 12-3 3 4. Title and Subtitle 5. Report Date Prediction and Prevention of Tripped Rollovers December 2012

6.

7. Author(s) 8. Performing Organization Report No. Gridsada Phanomchoeng and Rajesh Rajamani

9. Performing Organization Name and Address 10. Project/Task/Work Unit No.

Department of Mechanical EngineeringUniversity of Minnesota111 Church Street SEMinneapolis, MN 55455

CTS Project #201101911. Contract (C) or Grant (G) No.

12. Sponsoring Organization Name and Address 13. Type of Report and Period Covered Intelligent Transportation Systems InstituteCenter for Transportation StudiesUniversity of Minnesota200 Transportation and Safety Building511 Washington Ave. SEMinneapolis, MN 55455

Final Report14. Sponsoring Agency Code

15. Supplementary Notes http://www.its.umn.edu/Publications/ResearchReports/16. Abstract (Limit: 250 words)

Vehicle rollovers account for a significant fraction of highway traffic fatalities, causing more than 10,000deaths in the U.S. each year. While active rollover prevention systems have been developed by several automotivemanufacturers, the currently available systems address only untripped rollovers. This project focuses on thedevelopment of a new real-time rollover index that can detect both tripped and un-tripped rollovers.

A new methodology is developed for estimation of unknown inputs in a class of nonlinear dynamicsystems. The methodology is based on nonlinear observer design and dynamic model inversion to compute theunknown inputs from output measurements. The developed approach can enable observer design for a large classof differentiable nonlinear systems with a globally (or locally) bounded Jacobian.

The developed nonlinear observer is then applied for rollover index estimation. The rollover indexestimation algorithm is evaluated through simulations with an industry standard software, CARSIM, and withexperimental tests on a 1/8 th scaled vehicle. The simulation and experimental results show that the developednonlinear observer can reliably estimate vehicle states, unknown normal tire forces, and rollover index for

predicting both un-tripped and tripped rollovers. The final chapter of this report evaluates the feasibility of rollover prevention for tripped rollovers using currently available actuation systems on passenger sedans.

17. Document Analysis/Descriptors 18. Availability Statement Rollover accidents, Rolling, Estimation theory, Nonlinearsystems

No restrictions. Document available from: National Technical Information Services,Alexandria, Virginia 22312

19. Security Class (this report) 20. Security Class (this page) 21. No. of Pages 22. Price Unclassified Unclassified 95


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Prediction and Prevention of Tripped Rollovers

Final Report

Prepared by:

Gridsada PhanomchoengRajesh Rajamani

Department of Mechanical EngineeringUniversity of Minnesota

December 2012

Published by:

Intelligent Transportation Systems InstituteCenter for Transportation Studies

University of Minnesota200 Transportation and Safety Building

511 Washington Ave. S.E.Minneapolis, Minnesota 55455

The contents of this report reflect the views of the authors, who are responsible for the facts and the accuracy of theinformation presented herein. This document is disseminated under the sponsorship of the Department ofTransportation University Transportation Centers Program, in the interest of information exchange. The U.S.Government assumes no liability for the contents or use thereof. This report does not necessarily reflect the officialviews or policies of the University of Minnesota.

The authors, the University of Minnesota, and the U.S. Government do not endorse products or manufacturers. Anytrade or manufacturers’ names that may appear herein do so solely because they are considered essential to thisreport.


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Acknowledgments

The authors wish to acknowledge those who made this research possible. The study was funded by the Intelligent Transportation Systems (ITS) Institute, a program of the University ofMinnesota’s Center for Transportation Studies (CTS). Financial support was provided by theUnited States Department of Transportation’s Research and Innovative TechnologiesAdministration (RITA).


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Table of Contents

1. Introduction ............................................................................................................................ 1

1.1. Active Rollover Prevention .............................................................................................. 1

1.2. Rollover Index and Unknown Disturbance Inputs for Tripped Rollovers ....................... 2

1.3. Tripped Rollover Index Estimation .................................................................................. 5

2. The Bounded Jacobian Approach to Nonlinear Observer Desdign .................................. 7

2.1. Introduction ...................................................................................................................... 7

2.2. Problem Statement for Nonlinear Observer ..................................................................... 7

2.3. Mean Value Theorem for Bounded Jacobian Systems .................................................... 8

2.4. Nonlinear Observer ........................................................................................................ 12

2.5. Conclusions .................................................................................................................... 15

3. The Extended Bounded Jacobian Approach to Observer Design for Nonlinear Systemswith Nonlinear Measurement Equation.................................................................................... 17

3.1. Problem Statement for Nonlinear Observer ................................................................... 17

3.2. Mean Value Theorem for Bounded Jacobian Systems .................................................. 17

3.3. Nonlinear Observer ........................................................................................................ 18

3.4. Conclusions .................................................................................................................... 26

4. Novel Unknown Inputs Nonlinear Observer ..................................................................... 27

4.1. Introduction .................................................................................................................... 27

4.2. Problem Statement for Unknown Inputs Nonlinear Observer ....................................... 27

4.3. Unknown Input Estimation ............................................................................................ 28

4.3.1. Single Input Nonlinear Systems .............................................................................. 28

4.3.2. Multi-Input Nonlinear System ................................................................................. 29

4.4. Unknown Inputs Nonlinear Observer ............................................................................ 30

4.4.1. Nonlinear Observer ................................................................................................ 31

4.5. Conclusions .................................................................................................................... 32

5. Application of Nonlinear Observer to Rollover Index Estimation for Tripped and Un-Tripped Rollovers ....................................................................................................................... 33

5.1. Summary ........................................................................................................................ 33


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5.2. Introduction .................................................................................................................... 33

5.3. Vehicle Rollover Index .................................................................................................. 34

5.4. Vehicle Dynamics Model [22] ....................................................................................... 37

5.5. Observer Design for the Vehicle Problem ..................................................................... 40

5.5.1. Observer Design Using Corollary to Theorem 3 in Chapter 4 .............................. 41

5.6. Simulation and Simulation Results ................................................................................ 43

5.6.1. Simulation Setup ..................................................................................................... 43

5.6.2. Simulation Results ................................................................................................... 44

5.7. The Scaled Vehicle for Experiments .............................................................................. 46

5.7.1. Dynamic Similitude Analysis .................................................................................. 47

5.7.2. Experimental Set Up ............................................................................................... 51

5.7.3. Experimental Results .............................................................................................. 52

5.8. Alternate Rollover Index with Additional Measurements ............................................. 56

5.8.1. New Rollover Index for Tripped and Un-Tripped Rollovers .................................. 56

5.8.2. Derivation of Rollover Index .................................................................................. 58

5.8.3. Sensitivity Analysis to Mass Change....................................................................... 61

5.8.4. Simulation and Simulation Results ......................................................................... 62

5.8.5. Experimental Set Up ............................................................................................... 66

5.8.6. Experimental Results ............................................................................................. 66

6. Feasibility of Rollover Prevention in Tripped Rollovers .................................................. 71

6.1. Control Systems ............................................................................................................. 71

6.1.1. Rollover Prevention using Brake Torque Control .................................................. 71

6.1.2. Rollover Prevention using Semi-Active Suspension ............................................... 73

6.2. Simulations of Tripped Events during Straight Driving ................................................ 73

6.3. Simulations of Tripped Events during Cornering .......................................................... 76

6.4. Conclusions .................................................................................................................... 79

7. Conclusions ........................................................................................................................... 81

References ................................................... ................................................................................. 83


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List of Figures

Figure 1.1: Types of Rollovers ...................................................................................................... 1

Figure 1.2: NHTSA Rollover Record (http://www.safercar.gov) .................................................. 1

Figure 1.3: Un-Tripped Rollover Model ........................................................................................ 2

Figure 1.4: Un-Tripped and Tripped Rollover Model ................................................................... 3

Figure 5.1: Type of Rollover ....................................................................................................... 34

Figure 5.2: Vehicle Model for Un-Tripped Rollovers ................................................................. 35

Figure 5.3: Vehicle Model for Un-Tripped and Tripped Rollovers ............................................. 36

Figure 5.4: Four-Degrees of Freedom Vehicle Model. ................................................................ 37

Figure 5.5: Road Curvature .......................................................................................................... 43

Figure 5.6: Lateral Acceleration and Road Inputs ....................................................................... 43 Figure 5.7: Estimation of Right and Left Suspension Compressions .......................................... 44

Figure 5.8: Estimation of Right and Left Suspension Compression Rate ................................... 45

Figure 5.9: Roll Angle and Roll Rate Estimation. ....................................................................... 45

Figure 5.10: Normal Tire Forces Estimation, F zr and F z .............................................................. 46

Figure 5.11: Rollover Index Estimation ....................................................................................... 46

Figure 5.12: Scaled Test Vehicle: 1:8 (30.5 x 58.5 cm) .............................................................. 49

Figure 5.13: Microcontroller and Sensors ................................................................................... 51

Figure 5.14: The Scaled Vehicle Path .......................................................................................... 52

Figure 5.15: Longitudinal and Lateral Acceleration of the Scaled Vehicle ................................ 53

Figure 5.16: Estimation of Roll Rate ........................................................................................... 53

Figure 5.17: Estimation of Right Suspension Defection ( z s-z ur -l s si ϕ n /2 ) .................................... 54

Figure 5.18: Estimation of Roll Angle ......................................................................................... 55

Figure 5.19: Comparison of Rollover Indices of the Scaled Vehicle .......................................... 55

Figure 5.20: Four-Degrees of Freedom Vehicle Model ............................................................... 56

Figure 5.21: Lateral Vehicle Dynamics ....................................................................................... 57

Figure 5.22: Suspension Forces Direction ................................................................................... 58

Figure 5.23: Extra Accelerometer Locations ............................................................................... 60

Figure 5.24: Comparison of Rollover Indices.............................................................................. 64


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Figure 5.25: Rollover Indices with Step Steering Input ( δ =1.2 deg.) and Road Bump (z rl =0.15m) .................................................................................................................................................. 65

Figure 5.26: The Scaled Vehicle Path .......................................................................................... 66

Figure 5.27: Longitudinal and Lateral Acceleration of the Scaled Vehicle ................................ 67

Figure 5.28: Right and Left Vertical Acceleration of the Scaled Vehicle ................................... 68

Figure 5.29: Comparison of Rollover Indices of the Scaled Vehicle. ......................................... 69

Figure 5.30: Comparison of Rollover Indices of the Third Experiment ...................................... 69

Figure 6.1: Road Inputs on the Right and Left Sides ................................................................... 74

Figure 6.2: Vehicle Strikes a Bump during Driving Straight ...................................................... 74

Figure 6.3: Rollover Indices without any Control System and with a Semi-Active Suspension 75

Figure 6.4: Rollover Index of a Vehicle with an Automatic Brake Torque Control System ...... 76

Figure 6.5: Road Curvature .......................................................................................................... 76 Figure 6.6: Road Inputs on the Right and Left Sides ................................................................... 77

Figure 6.7: Vehicle Strikes a Bump during Cornering ................................................................ 77

Figure 6.8: Rollover Index of a Vehicle without any Control System ........................................ 78

Figure 6.9: Rollover Index with a Semi-Active Suspension System ........................................... 78

Figure 6.10: Rollover Index with a Brake Control System ......................................................... 78


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List of Tables

Table 5.1: Summary of Parameters Associated with the Vehicle Dynamics .............................. 48

Table 5.2: π Groups ..................................................................................................................... 49

Table 5.3: Vehicle Variables and Parameters .............................................................................. 50

Table 5.4: Comparison of π Groups ............................................................................................. 50


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Executive Summary

Vehicle rollovers account for a significant fraction of highway traffic fatalities, causing morethan 10,000 deaths in the U.S. each year. An active rollover prevention system is a vehiclestability control system that prevents rollovers. In such systems, reliable detection of the dangerof an impending rollover is necessary. A rollover index is a real-time indicator used for this

purpose. A traditional rollover index can detect only untripped rollovers that happen due to highlateral acceleration from sharp turns. It cannot detect tripped rollovers that happen due totripping from external inputs such as forces when a vehicle strikes a curb or a road bump.

While active rollover prevention systems have been developed by several automotivemanufacturers, the currently available systems address only untripped rollovers. This projectfocuses on the development of a new rollover index that can detect both tripped and untrippedrollovers.

A methodology is developed for estimation of unknown inputs in a class of nonlinear systems.The methodology is based on nonlinear observer design and dynamic model inversion tocompute the unknown inputs from output measurements. The observer design utilizes the meanvalue theorem to express the nonlinear estimation error dynamics as a convex combination ofknown matrices with time varying coefficients. The observer gains are then obtained by solvinglinear matrix inequalities (LMIs). The developed approach can enable observer design for a largeclass of differentiable nonlinear systems with a globally (or locally) bounded Jacobian.

The developed nonlinear observer is then applied for rollover index estimation. The rolloverindex estimation algorithm is evaluated through simulations with an industry standard software,CARSIM, and with experimental tests on a 1/8 th scaled vehicle. In order to verify that the scaledvehicle experiments can represent a full-sized vehicle, the Buckingham π theorem is used to

show dynamic similarity. The simulation and experimental results show that the developednonlinear observer can reliably estimate vehicle states, unknown normal tire forces, and rolloverindex for predicting both un-tripped and tripped rollovers.

The final chapter of this report evaluates the feasibility of rollover prevention for trippedrollovers using currently available actuation systems on passenger sedans. Active brake torquedistribution and semi-active suspensions are evaluated for their ability to prevent trippedrollovers using the rollover index estimation algorithm developed in this project. Simulationswith CARSIM show that the rollover index can play a key role in preventing a rollover with boththe brake torque control and semi-active suspension actuation systems.


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Chapter 1. Introduction

1.1. Active Rollover Prevention

Rollovers occur in one of two ways, namely tripped or un-tripped [1]. The two types of rolloversare shown in Figure 1.1. A tripped rollover happens due to tripping from external inputs. Anexample of this rollover happens when a vehicle leaves the roadway and slides sideways, diggingits tires into soft soil or striking an object such as a curb or guardrail. On the other hand, an un-tripped rollover happens due to high lateral acceleration from a sharp turn and not due to externaltripping. An example of un-tripped rollover is when a vehicle makes a collision avoidancemaneuver or a cornering maneuver with high speed.

Rollover from vertical inputs Roll from lateral inputs

a) Tripped Rollover b) Un-Tripped Rollover

Figure 1.1: Types of Rollovers

Rollover accidents are dangerous. According to NHTSA’s records ( http://www.safercar.gov ),although there were nearly 11 million crashes in 2002, only 3% involved a rollover. However,there were more than 10,000 deaths in rollover crashes in 2002. Thus, rollovers caused nearly33% of all deaths from passenger vehicle crashes. In addition, NHTSA data also shows that 95%

of single-vehicle rollovers are tripped while un-tripped rollover occurs less than 5% of the time.

Figure 1.2: NHTSA Rollover Record (http://www.safercar.gov)

Several types of actuation systems can be used for rollover prevention. The differential brakingsystem has received the most attention from researchers. It is used to prevent rollovers byreducing the yaw rate of the vehicle and the speed of the vehicle. By reducing yaw rate andspeed, the vehicle propensity to rollover is reduced. Also, steer-by-wire systems and activesuspensions can potentially be used to prevent rollovers.

Active rollover prevention systems have already by developed by several automotivemanufacturers and are based on modifications of the electronic stability control systems. These


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systems utilize lateral accelerometers to detect rollover and can detect and prevent onlyuntripped rollovers [2]. There are no assistance systems available to prevent tripped rollovers.

1.2. Rollover Index and Unknown Disturbance Inputs for Tripped Rollovers

The quick detection of the danger of a vehicle rollover is necessary for initiating a rollover prevention action. To detect vehicle rollover, the concept of a static rollover threshold or thestatic stability factor (SSF) [3] was studied and used first, but it is not effective for dynamicsituations. After that the concept of a rollover index was introduced. Rollover Index is a real timevariable used to detect wheel lift off conditions. Many researchers have tried to develop arollover index to more accurately predict vehicle rollover for un-tripped rollovers. Reference [4]has used the concept of a rollover index based on Time-To-Rollover (TTR) to estimate the timeuntil rollover occurs. References [2], [5], and [6] have described a rollover index using a model-

based roll angle estimator. Reference [7] has combined a rollover index with influential factorssuch as the vehicle’s center of gravity and energy of rollover. Even though there are many typesof rollover indices, they are derived from the same basic model as shown in Figure 1.3. The basicconcept of the rollover index is described by equation (1.1):

,where

1 1 (1.1)

and are the right and left vertical tire force of the vehicle respectively, 1 1 is

unsprung mass, is spung m

0 1ass, is lateral acceleration, is roll angle, and is roll rate. A

vehicle is considered to roll over when equals or . This is when wheel lift-off occurs. Forexample, when , and the left wheels lift off. Likewise, when 0, andthe right wheels lift off. It should be noted that when a vehicle is traveling straight, equals to

and [22].

0

Figure 1.3: Un-Tripped Rollover Model

The definition of in equation (1.1) cannot be implemented in real-time because the vertical tireforces and cannot be measured. Using the 1-degr

ee of freedom m

odel in Figure 1.3, the

summation and difference of tire forces and can be estimated for theuntripped scenario. An implementable version of the rollover index can then be calculated in


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terms of and . Such an example of a traditional rollover index calculated using a one degreeof freedom is shown below in equation (1.2) [22]:

2

2

where , is c.g. height, is unsprung mass,acceleration, and is roll angle.

(1.2)

is spung mass, is lateralThis type of rollover index is used for detection un-tripped rollovers only. It is a function oflateral acceleration and roll ang le as shown in equation (1.2). Some papers have proposed amodification of the above rollover index that uses only lateral acceleration [8], [9], since rollangle is expensive to measure. The stability control with this rollover index may arbitrarilyreduce lateral acceleration capability of the vehicle. Also, it still fails to detect rollovers whenrollovers are induced by vertical road inputs or other external inputs.

In order to detect tripped rollovers, which happens due to tripping from external inputs, a newrollover index should include the influence of road and other external inputs. Figure 1.4 shows avehicle rollover model that includes the influence of road inputs,

and

, from, and an unknown

lateral force input, , at an arbitrary height, the roll center. The figure also showsthe normal forces on the tires, and .

Figure 1.4: Un-Tripped and Tripped Rollover Model


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To derive a rollover index in th is case, multi-degrees of freedom in the model are needed.When the influence of road inputs is in cluded, the suspension forces are defined by

2 2 , spension stiffness, 2 where is su 2 (1.3)

mass positioand the vertical displacem

ns, and is sprung ments due to road inputs.

(1.4)

is suspension damping,in

and are left and right unsprungass position. cludes the static displacement due to weight

The difference and the summ are given by ation of and

2 2

With moment balan ce at the ro ll cen ter,

. (1.5), (1.6)

is given by 2Also, vertical force balance yields

(1.7)

Therefore, an exam ple of the rollov

model of Figure 1.4 is given by

2er index fo

2 r tripped rollovers com

(1.8) puted directly from

the

2 where left and right uns

2 (1.9)

prung m

ass position2 2

s,

and , depend on road inputs,

The equations of motion of and are given by

and .

m (1.11)

where , (1.10)

is the vertical tire stiffness,ass, respectively. and are right unsprung mass and left unsprung

In order to compute the rollover index in equation (1.9), many variables need to be measured.However, some variables such as unknown road inputs, vertical displacements of unsprungmasses and sprung mass, and the unknown lateral force input cannot be directly measured by


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sensors. Therefore, it is necessary to develop an approach for estimating the new rollover indexwithout knowing these variables.

1.3. Tripped Rollover Index Estimation

In order to predict tripped and un-tripped rollovers as described in section 1.2, many variablesneed to be measured. However, some variables such as unknown road inputs, verticaldisplacements of unsprung masses and sprung mass, and the unknown lateral force input cannot

be directly measured by sensors. Therefore, algorithms to estimate the rollover index for trippedand un-tripped rollovers are developed.

Two different algorithms are considered in this project based on the types of sensors available:

1) Using accelerometers, gyroscope, and suspension compression measurement2) Using accelerometers, and roll angle measurement

The first algorithm is based on an approach to estimate unknown disturbance inputs in a

nonlinear system using dynamic model inversion and a modified version of the mean valuetheorem. In this case, the vehicle dynamics involve both complex nonlinearities and unknowndisturbance inputs. The dynamics of a vehicle in this case can be written into the state space formshown in equation (1.12).

, , , (1.12)where are the unknown inputs,are the outp ∈ u are the known control inputs,t measurem ents. Ψ

∈ ,

m

∈ ∈

atrices. The functions

, : → ,and

,

∈ , and

∈ ∈

Ψ: → are nonlinear

and ∈ are appropriate.

To deal with this type of system, a new unknown inputs nonlinear observer to estimate bothunknown disturbance inputs and state variables has been developed. The developed observerdesign for this problem is presented in chapters 2, 3 and 4. Then, the application of the unknowninputs nonlinear observer for rollover prevention and corresponding experimental results will be

presented in chapter 5.

The second algorithm relies on an algebraic formulation of the new rollover index. The newrollover index utilizes roll angle measurement and vertical accelerometers in addition to a lateralaccelerometer and is able to predict rollover in spite of unknown external inputs acting on thesystem. The details of this new rollover index are also presented in chapter 5.

Chapter 6 of the report focuses on showing the feasibility of rollover prevention using thedeveloped rollover prediction algorithm from previous chapters and using either brake torquedistribution or semi-active suspension actuation systems.

̅


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Chapter 2. The Bounded Jacobian Approach to NonlinearObserver Desdign

2.1. Introduction

This chapter presents a new observer design technique for a nonlinear system with a globally (orlocally) bounded Jacobian. The developed approach can enable observer design for a large classof differentiable nonlinear systems. Its advantage is that it enables easy observer design for amuch wider range of operating conditions compared to linear or Lipschitz observer designmethods. A major limitation of the existing results for Lipschitz nonlinear systems is that theywork only for adequately small values of the Lipschitz constant. When the equivalent Lipschitzconstant has to be chosen large due to the inherent non-Lipschitz nature of the nonlinearity (suchas in the case of aerodynamic drag force in vehicle systems), most existing observer designresults fail to provide a solution. This section develops a solution methodology that works wellwithout requiring a small Lipschitz constant bound for the nonlinear function. The basic idea in

this section is to use the mean value theorem (McLeod (1965) and Korobkov (2001)) to expressthe nonlinear error dynamics as a convex combination of values of the derivatives of thenonlinear function. The observer gain guaranteeing the convergence of the proposed observercan then be easily computed by LMIs.

2.2. Problem Statement for Nonlinear Observer

This section presents an efficient methodology for designing observers for the class of nonlinearsystems described by

where ∈ is the state vecto ∈ r , measurem Φ ,ent vector. and, and ∈ (2.1)

Φ: → ,: ∈ → is the input vector, and

atrices. The functions is the output

are appropriate m are nonlinear. In addition ∈ al,Φ is assumed to bedifferentiable.

The observer will be ass umed to be of the form,

Φ, . (2.2)


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The estimation error introductions ar e then seen to be given by

Φ Φ ΦwhereLet the Lyapunov function

,

Φ ΦΦ.(2.3)

candidate for observer design be defined as

where 0 and ∈ . (2.4)

Then, its derivative is

Φ Φ . (2.5)

2.3. Mean Value Theorem for Bounded Jacobian Systems

In this sub-section, we present mathematical tools which are used subsequently to develop thedesign for the observer gain in the next section. First, we present the mean value theorem forscalars and the mean value theorem for vector functions. Then, we define the canonical basis forwriting a vector function with a composition form. Lastly, we present a new modified form ofthe mean value theorem for vector functions.

Lemma 1: Scalar Mean Value Theorem

Let

,

∈ : ,→, be a function continuous on

, ⊂ and differentiable on

,. For

, there exist num bers such that

∈ , The equation (2.6) can also be rewritten as

. (2.6)

where

, 0, ,

proof of this lemm

,Lemma 2: Mean Value Theorem∈ , and

and

are parameters that va ry with the value of and

(2.7)

. Thea is presented in [10].

for a Vector Function, [11]


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Let be a function continuous onof the set : , → with a Lipschitz continuous gradientsuch that ,∈ ,

and differentiable on a convex hull, ∈ , ∈However, we cannot directly use the m ean

parameter that continuously ch anges with the values of value theorem

. For

of e.

, there exists

(2.8)

quation (2.8), since is a varying. Thus is an unknown

and changing matrix. We need to modify the mean value theorem before it can b e utilized.

Canonical B

Lemma 3: asis, [12]

and Let a vector function be defined by:

: → . (2.9)Then,

where , ,…, Let the cano

: is the i th component of andnical b→

asis of the vectorial space

for all ∈ (2.10)

1.

The vectorial space

is generated by the canonical basis| 0,…,0,1,0,…,0,

defined by:

1,2,…,. (2.11)

. Therefore,

can be written as:

. (2.12) Now, we are ready to state and prove a modified form of the mean value theorem for a vectorfunction.

Theorem 1: Modified M ean Value Theorem for a Vector Fun

Let be a function continuous on. For , there exist and

ction

and differentiable on convex hull ofthe set forthat:

,: →

, ∈ , , ∈

1,…, and

1 such

, , , , , 1(2.13)

, 0,


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where 1) ⁄ and2) and ⁄Proof:

Lemma 2 shows that

for∀ ∈ , .

⁄ ⁄ … ⁄ ⁄ ⁄ … ⁄ .Lemma 1 shows that each derivative function canvalues of the derivative of the function. Hence, the

⁄⋮ ⁄⋮ …⋮ ⋮ ⁄ be replaced with a convex com

(2.14)

derivative function,replaced with ⁄ bination of 2, can be

Ω

, 0, ,where and 1

(2.15)

To satisfy lemma 1, the values ofΩ Ω,Ω,…,Ω ,,…,

and

. ,Ω, ∈ ,.

need to be chosen such that

Ω

Ω⁄ ⁄ ,and

Note: One can easily show that if either

.

, then there are no and

(2.16)

∀ ∈ , , 0 and 1. ⁄ or

⁄ for

that will satisfy equation (2.15) with the constraintsThen, the equation (2.15) can be rewritten as

where

, (2.17)

vary with the value of

, 0, ⁄

and

and

⁄. Note:

, are parameters that

. Hence, the equation (2.14) can be rewritten as


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or

, (2.22)

where ⁄ , 0, . , , ⁄ and

2.4. Nonlinear Observer

Theorem 2: Bounded Jacobian Observer for General Problems [13]

For the class of systems and observer forms described in equations (2.1) and (2.2), if an observergain matrix can be chosen such that

0 0∀ 1,…, ,and

(2.23)

where,

1)

2) 3) ⁄ Φ

andand

4) is the state scaland Φ ⁄ ,

,

then this cho ice of leads to asym

ptotically stable estim

, being dimension of the state vector,,ates by the observer (2.2) for the system

(2.1).

Proof: The derivative of

the Lyapunov function is

The nonlinear term

Φ

Φ

. (2.24)

s can be rewritten using theorem 1 as

Φ Φ Φ

, ,, , , (2.25)

, 0, 1.


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To simplify the form of the final result, we need to scalegeneral problem, if all the terms in Φ ⁄ ∑ are to be considered a,factors, is computed by , to one. In thend not zero, then the scaling

,Rewrite equation (2.25) as

, , ∑,, 1 (2.26)

Φ , , , , ,, (2.27)

where, , 0, 1⁄ , , 1)

2) , Then, the derivative of the Lyapunov function becom

⁄ andand ⁄ .

e

, s

,

, , , , , ,

(2.28)

, ,

, ,

, , , (2.29)

, Since ∑,, 1, equation (2.29) can be rewritten as

̅ ̅ ̅ ̅ ̅ ̅ ̅

̅ ̅

̅ ̅ ̅ ̅

̅ ̅

̅ ̅ ̅ ̅


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, , ,

,,

.,

(2.30)

, ,

, ,

,

.Hence we need , (2.31)

0 (2.32)where

because

equations (2.33) and (2.34).

, 0. However, it is not possible to directly solve equation (2.32) for and , are time varying coefficients. Hence, we transform equation (2.32) to

for

∀ 1,…, ,and ∀ 1.

(2.33)

Then if equation (2.33

(2.34)

) and (2.34) are satisfied, equation (2.32) will automatically be satisfied.

Corollary to Theorem 2: Bounded Jacobian Observer for Sparse Problems

For the class of systems and observer forms described in equations (2.1) and (2.2), if an observer

gain matrix can be chosen such that

0 0 (2.35)

∀ 1,…, ,and

̅

̅ ̅ ̅

̅

̅ ̅ ̅ ̅ ̅


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where,

1)

2) ⁄ Φ

is the state scali

andand Φ ⁄ ,

Φ ⁄ ,

3) 3) being dimension of the state vector, being the num

,

ber of terms in that equals zero,4) andthen th .is choice of leads to asymptotically stable estimates by the observer (2.2) for the system

(2.1).

Proof: The proof of the Corollary follows along the same lines as the proof of theorem 2, exceptfor the definition of the scaling factor .

In the gen

eral problem, if all of terms in Φ

. However, if in som are not zero, this les

e problem, there exist Φ

s than . We need to define a new scaling factor,

⁄ ∑ .⁄

,

,

0en

, then

∑ ,,

, wherecomplete the proof.

2.5. Conclusions

, ⁄

(2.36)

is number of terms in Φ that equals zero. Now, we use instead of to

In this chapter, a new observer design technique is developed for a nonlinear system with aglobally (or locally) bounded Jacobian. The approach is developed in order to deal withdifferentiable nonlinear systems. The observer gains in the developed approach can be obtained

by solving LMIs.

̅ ̅ ̅ ̅

̅ ̅ ̅


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Chapter 3. The Extended Bounded Jacobian Approach toObserver Design for Nonlinear Systems with Nonlinear

Measurement Equation

This chapter extends the result in chapter 2 and presents how the technique can be applied for anonlinear system with a nonlinear measurement model.

3.1. Problem Statement for Nonlinear Observer

This chapter presents an efficient methodology for designing observers for the class of nonlinearsystems described by

where

,m

,easurem

functions ∈ent vecto Φ (3.1)Ψ

is th e state vecto

: r.Φ ∈ ∈ → , Ψaddition, Φ Ψ ∈ and are assumThe observer will be assumed to be of

r,

∈ is the input vector, and

and: are apped to be differentiable.→ , andthe form

,tput

rop: → is the ou

riate matrices. Theare nonlinear. In

The estimation error introductions ar

Φ

Ψ

e then seen to be given by. , (3.2)

where , Let the Lyapunov function

Φ Φcandidate for observer

Φ Φ

, and Ψ Ψdesign be defined as

ΨΨ (3.3)

.

where . 0

∈ and

(3.4)

. Then, its derivative is

Φ

Φ

Ψ

Ψ

. (3.5)

3.2. Mean Value Theorem for Bounded Jacobian Systems

The mathematical tools which are used subsequently to develop the observer gain in the nextsection are the same as one in the chapter 2 section 2.3. So, we will not repeat them again.


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The nonlinear terms can be rewritten using theorem 1 as

Φ Φ Φ

,

, , , ,

,(3.8)

Ψ Ψ Ψ , 0, 1

, , ,To simplify the form of the f

,

(3.9)

∑ inal re,

sult, we need to scale 0,

to one. In the general problem, if all the te∑,, 1 rms in Φ⁄,, and and

Ψ⁄

are to be considered and

, not zero, then the scaling factors, and are computed by

, , , , ∑ 1 , , , ∑, Rewrite equation (3.8) and (3.9) as

, 1 (3.10)

Φ ,, ,, ,, 0, 1⁄ , ,

(3.11)

1

, ,Ψ ,, ,

, , , 0, 1⁄ , , , 1

(3.12)

̅ ̅ ̅ ̅ ̅ ̅

̅ ̅ ̅ ̅ ̅


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where,

1) and2) 3)

⁄

⁄ ,,

4) ,Then, the derivative of the Lyapunov function becom ⁄ and

and , and ⁄

.

es

, , , , ,, , , (3.13) , , , ,

, , , ,

̅ ̅ ̅ ̅ ̅ ̅

̅ ̅ ̅ ̅ ̅ ̅ ̅ ̅ ̅ ̅ ̅ ̅


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̅ ̅

Since ∑,, 1 ,, equation (3.13) can be rewritten as

,

, , , , , , , ,

, , ,

. (3.14)

, , , , , , , , ,

, ,

̅ ̅ ̅ ̅ ̅

̅ ̅ ̅ ̅ ̅ ̅ ̅ ̅ ̅ ̅

̅ ̅ ̅


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, ,

, ,

,

, , , ,

, , ., (3.15)

, ,

,

,

, , Hence we need

, , where

0 because

,

equations (3.17) and (3.18).

(3.16)

,. However, it is not possible to directly solve equation (3.16) for and

e varying coefficients. Hen 0 are tim ce, we transform equation (3.16) to , , , , 0 ,

, ,

(3.17)

,

̅ ̅ ̅ ̅ ̅ ̅ ̅ ̅ ̅

̅ ̅ ̅ ̅ ̅ ̅ ̅

̅ ̅ ̅ ̅

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, , ,

0

(3.21)

, , , ,

, (3.22) 0

However, it is still not possible to

because , solve equation (3.19) and (3.20) for

and ,and are time varying coefficients. Hence, to overcom e this problem , we

transform the problem in equations (3.19) and (3.20) to equations (3.23)-(3.26) which do notinvolve .

0, (3.24)

for

(3.23)

0,(3.25)

If equations (3.23)-(3.26) are satisfied, then∀ 1,…, , ∀ 1 ,…,, and∀ 1,…,.

(3.26)

equation (3.16) is automatically satisfied.

Corollary to Theorem 3: Bounded Jacobian Observer for Sparse Problems

For the class of systems and observer forms described in equations (3.1) and (3.2), if an observergain matrix can be chosen such that

̅ ̅ ̅ ̅

̅ ̅ ̅

̅ ̅ ̅ ̅ ̅ ̅ ̅ ̅ ̅ ̅ ̅


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0

0 (3.27)

Where,∀ 1,…, ,∀ 0 and∀ 1,…

1)

2) 3)

⁄ Φ

is the state scaling factor, and ⁄

being the number of term and Φ , ,

4)

5)

⁄ 6)

being dimension of the state vector,

7)

s in Φ that equals zero,and

Ψ

⁄ and

,

Ψ

⁄ being the number of term and, is the state scaling factos in Ψ

,,

8) and then th

⁄ r, being dimension of the output vector, that equals zero,,is choice of leads to asymptotically stable estimates by the observer (3.2) for the system

(3.1).

Proof: The proof of the Corollary follo ws along the same lines as the proof of theorem 3, except

for the definition of the scaling factor and .

In the gen er al problem, if all of terms in Φ are not zero, then∑ or if all of term

s in Ψ are not zero, then⁄ , . However, in som ,, ∑,

,, e problems, if there exist Φ or Ψ , then

is less than

⁄ ∑ , as f ollows:

orand ∑, new scaling factors, ⁄ ⁄ 0 is less th at 0 . We can then define

,

, ,

(3.28)

, (3.29)where ⁄ is the number of terms in Φ that equals zero and is the number of terms inΨ that equals zero. Now, the use of and instead of and can be used tocomplete the proof. ⁄

̅ ̅ ̅ ̅ ̅ ̅ ̅ ̅

̅ ̅ ̅ ̅ ̅ ̅ ̅ ̅

̅ ̅ ̅ ̅

̅ ̅


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3.4. Conclusions

In this chapter, a new observer design technique is developed for a nonlinear system with alocally or globally bounded Jacobian and a nonlinear output measurement equation. The resultsdeveloped in chapters 2 and 3 will be extended to address disturbance estimation for nonlinearsystems in chapter 4.


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Chapter 4. Novel Unknown Inputs Nonlinear Observer

4.1. Introduction

This chapter extends the result in chapter 2 and chapter 3 and presents an observer designtechnique for an unknown inputs nonlinear system with a globally (or locally) bounded Jacobian.This observer can applied for an unknown inputs nonlinear system with nonlinear measurementmodel. The approach utilized is to use measurements and state estimates to express the unknowninputs and use the mean value theorem to express the nonlinear error dynamics as a convexcombination of known matrices with time varying coefficients. The observer gains are thenobtained by solving linear matrix inequalities. The developed approach can enable observerdesign for a large class of differentiable nonlinear systems with a globally (or locally) boundedJacobian. The developed theory is used successfully in the design of observers for vehiclesystems involving complex nonlinearities and unknown inputs. The use of the observer designtechnique is illustrated for estimation of roll angle and rollover index in chapter 5.

4.2. Problem Statement for Unknown Inputs Nonlinear Observer

This section presents an efficient methodology for designing observers for the class of nonlinearsystems described by

, (4.1) ∈

(4.2)Ψ (4.3)

where are the known control inputs, are the unknown inputs,are the outpu t measurem

∈ matrices. The functions

ents.

, :∈ ,

→ ∈ ,

,and

∈Ψ

: →

, and ∈ and

, ∈

are app

are nonlinear. In addition,

ropriate ∈ and Ψ are assumed to be differentiable nonlinear functions with globally (or locally)

bounded Jacobian.

The objective of this estimation problem is to quantitatively estimate the unknown inputsand to es timate the state vector , given the output measurements and . Theapproach to estimate unknown inputs will first be described in the section 4.3 and subsequentlythe nonlinear observer design technique will be described in the section 4.4.

The overall approach to the estimation problem is as follows:

1) The unknown inputs are estimated assuming that full state measurement is available.The algebraic relation between the unknown inputs, the states and outputs isdeveloped.

2) Using the algebraic relation between the unknown inputs and states, a modified stateintroductions equations are developed which do not depend on the unknown inputs.

3) Stable observer design for the new modified nonlinear introductions ensures that thestate and unknown inputs can both be estimated.

̅


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4.3. Unknown Input Estimation

To make the presentation easy to follow, we will consider single input nonlinear systems first.Then, we will extend the results to multi-input nonlinear systems.

4.3.1. Single Input Nonlinear Systems

Consider equations (4.1)-(4.3) whennumber of output measurements,

tive degree from to

).

Let the rela be

. Hence

1( is number of unknown inputs, ) and ( is

0, 0 Define the relative degree from the nonlinear functionoutput

0 . 1 (4.4) (4.5)words,

, must be differentiated before the nonlinear function

is defined as a whole number such that

, to as the number of times the

is encountered. In order

, 0, ,0 Theorem 4: Let the relative degrees from the input

0. (4.6) 1 (4.7) and from the nonlinearity be such that. Then the estim

ated unknown input is given by

where we assume that the state variables, , are kn

own, and the f ilte

,red der

, are given by

(4.8)

1 1ivatives of the output

0 0 0 1 ⋮ ⋱ 0 1 1 ⋱ ⋱ ⋱ 01 ⋮ 1 ⋮ . 1 … fr

Proof : Since the relative degree of the system

1

om the outp ut to the unknown input

f

(4.9)

is , it

ollows that

(4.10)Since

, , it therefore follows that th e unknown input is given by (4.11)

̅ ̅

̂ ̅ ̅

̅ ̅ ̅ ̅ ̅ ,


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̂Since is measured but the th derivative of the output is not unavailable, we therefore usethe following estim ator for the unknown input.

ven in equatiwhere is obtained from the th order filter gi on (4.9). Then, as the filter

,constant

(4.12)

→ 0,4.3.2. Multi-Input

→ as → ∞ . Nonlinear System

Consider equations (4.1)-(4.3) whennumber of output measurements, ).

Let the vector relative degree of the system

1( is number of unknown inputs) and 1 ( is

from the outputs to the input be …

,

,…,. Let the relative deg

r ee f

and . Assume that

follows that

rom each of

, the outpu

,…,ts to the nonlin

earity be

and

. It

Hence

, , 1, (4.13) ⋮

⋮

⋮ , ⋮ .

It therefore follows that the unknown inputs are given by (4.14)

⋮

⋮

⋮

(4.15) ⋮ ⋮ , .

̅

̂

̅ ̅ ̅ ̅

̅ ̅

̅ ̅ ̅ ̅ ̅


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Theorem 5: The estim ated unknown inputs are given by

⋮

⋮

,where we assume that the state variables,

⋮ ⋮ ⋮ (4.16)

1 0 0 0, are known and

1 1 ⋮ 1 1

⋱ ⋱ ⋱ 0 ⋮ 1

⋱ 1

0 ⋮ ,

, 12Proof: Follows along the sam … e lines as the proof for Theorem 4. 1

(4.17)

1

4.4. Unknown Inputs Nonlinear Observer

In section 4.4, we have shown that the estimated unknown inputs, , can be estimated by usingequation (4.16). So, we can substitute the unknown inputs, , by using equation (4.16). Then, thenew dynamic equation for the nonlinear system

is given by

, ,

Rearrange equation (4.18), as

⋮ (4.18)

⋮ ⋮ ⋮ Φ , ,

Φ ⋮ ⋮

,

, , (4.19)

⋮ ,, ⋮

̂ ̂ ̂

̅ ̅ ̅ ̅ ̅

̂

̅ ̅ ̅ ̅ ̅ ̅ ̅

̅ ̅ ̅

̅ ̅ ̅ ̅ ̅ ̅ ̅ ̅


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3) being the number of term

4)

5)

6)

⁄

is the state scaling factor,

being dime nsion of the state vector,

7)

Φ that equals zero,

Ψ ⁄

s in

and

, is the s

,

andandtate scaling facto ⁄

Ψ

,

being the number of term8) and s in Ψ ⁄

,,

i

r being dimension of the output vector,

then this cho ce of leads to asymptotically stable es that equals zero,,

timates by the observer (4.21) for thesystem (4.20).

4.5. Conclusions

In this chapter, a new observer design technique is developed for an unknown inputs nonlinearsystem with a locally or globally bounded Jacobian. The approach is developed in order to dealwith differentiable nonlinear systems with unknown inputs. The observer gains can be obtained

by solving LMIs. The use of the observer design technique is illustrated for estimation of rollangle and rollover index in chapter 5.

̅ ̅ ̅ ̅ ̅ ̅ ̅ ̅


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Chapter 5. Application of Nonlinear Observer to RolloverIndex Estimation for Tripped and Un-Tripped

Rollovers

In order to predict tripped and un-tripped rollovers, as described in chapter 1, many variablesneed to be known. However, some variables such as unknown road inputs, vertical displacementsof unsprung masses and sprung mass, and the unknown lateral force input cannot be directlymeasured by sensors. Therefore, two different algorithms to estimate the rollover index fortripped and un-tripped rollovers have been developed.

The first algorithm is an approach to estimate unknown disturbance inputs in a nonlinear systemusing dynamic model inversion and a modified version of the mean value theorem, as describedin chapter 4. The developed theory is used to estimate vertical tire forces and predict trippedrollovers in situations involving road bumps, potholes, and lateral unknown force inputs.

The second algorithm utilizes a new algebraic formulation of the new rollover index. Thisalgorithm is simple and convenient but can be used only if the roll angle and verticalaccelerations at multiple locations on the vehicle body are available as measurements.

5.1. Summary

Accurate detection of the danger of an impending rollover is necessary for active vehicle rollover prevention systems. A real-time rollover index is an indicator used for this purpose. A traditionalrollover index utilizes lateral acceleration measurements and can detect only un-tripped rolloversthat happen due to high lateral acceleration from a sharp turn. It cannot detect tripped rolloversthat happen due to tripping from external inputs such as forces when a vehicle strikes a curb or a

road bump. Therefore, this project develops a new rollover index that can detect both tripped andun-tripped rollovers.

5.2. Introduction

Vehicles with increased dimensions and weights are known to be at higher risk of rollover. Normally, rollovers occur in one of two ways, tripped and un-tripped [1]. The two types ofrollovers are shown in Figure 5.1. A tripped rollover happens due to tripping from externalinputs. An example of this rollover happens when a vehicle leaves the roadway and slidessideways, digging its tires into soft soil or striking an object such as a curb or guardrail. An un-tripped rollover, on the other hand, happens due to high lateral acceleration from a sharp turn andnot due to external inputs. An example of an un-tripped rollover is when a vehicle makes a sharpcollision avoidance lane change maneuver or a cornering maneuver at high speed, andconsequently rolls over.


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a) Tripped Rollver b) Un-tripped Rollover

Figure 5.1: Type of Rollover

An active rollover prevention system is a vehicle stability control system that prevents vehiclesfrom un-tripped rollovers. It has been developed by several automotive manufacturers [15], [2],e.g. Ford and Volvo. To the best of the authors’ knowledge, there are no assistance systemscurrently available that directly address tripped rollovers. Several types of actuation systems can

be used in rollover prevention. A differential braking system has received the most attention

from researchers [16], [17], [18] and is used for preventing rollovers by reducing the yaw rate ofa vehicle and its speed. Also, steer-by-wire and active suspension systems can be potentiallyused to prevent rollovers [7], [19].

In order to make these systems effective in their tasks, accurate detection of the danger of un-tripped and tripped vehicle rollovers is necessary [2]. To detect a vehicle rollover, manyresearchers have developed a real-time index that provides an indication of the danger ofrollover. However, they have focused on developing an indicator only for un-tripped rollovers.There are no currently published papers that have studied how to detect tripped vehicle rolloverswith unknown external inputs.

In this chapter, we present a method of estimating vehicle states based on a nonlinear vehiclemodel. The estimated vehicle states are used to calculate unknown normal tire forces, and arollover index that can detect both un-tripped and tripped rollovers. The method is suitable for alarge range of operating conditions. To begin with, we will introduce the vehicle rollover indexin section 5.3. Then the vehicle dynamic model is presented in section 5.4. In section 5.5, weapply the new observer design of chapter 4 to the vehicle problem. After that, in section 5.6, weevaluate the developed estimation algorithm by implementing it in CARSIM, an industrystandard vehicle dynamics simulation software. Moreover, we evaluate the estimation algorithmwith experimental tests on a 1/8 th scaled vehicle in section 5.7. An alternate rollover index withadditional measurements will be presented in section 5.8. Finally, the conclusions are presentedin section 5.9.

5.3. Vehicle Rollover Index

Initially, the concept of a static rollover threshold called the static stability factor (SSF) [3], [20]was studied to detect vehicle rollovers. However, the SSF by itself is not adequate for rollover

prediction in dynamic situations. After that the concept of a rollover index has been introduced.A rollover index has also been known by other names such as Roll Safety Factor (RSF) and LoadTransfer Ratio (LTR). A rollover index is a real-time variable used for rollover prevention. Asimple method of defining the rollover index is based on the use of the real-time difference


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in vertical tire loads between left and right sides of the vehicle, as shown in Figure 5-2. Theformula of the rollover index is

described by equation (5.1) [21].

where and , 1 1

(5.1)

( ) orfrom approachingequals to and

are the right and left vertical tire hen( forces of a vehicle respectively. W), the left or right wheels lift off. Thus should be prevented or . It should be noted that when a vehicle is traveling straight, .

Figure 5.2: Vehicle Model for Un-Tripped Rollovers

The definition of in equation (5.1) cannot be easily implemented in real-time because thevertical tire forces and cannot be directly measured. Thus

derive an estimation of the rollover index based on a 1-degree of freedFigure 5.2. The 1-degree of freedom is the roll anglea traditional rollover index calculated using a one degree of freedom(5.2):

, many researchers have tried to

om model as shown inof the vehicle body. Such an example of

is shown below in equation

2 where

2mass,

This type of rollover index is used for detecti

, is c.g. height, (5.2)

is track width,

is unsprung mass, is spungis lateral acceleration, and is roll angle.

ng un-tripped rollovers onl

y. It is a function oflateral acceleration and roll angle. Some papers have proposed a rollover index that uses only

lateral acceleration [8], [9] since roll angle is expensive to measure. The stability control withthis rollover index may arbitrarily reduce the lateral acceleration capability of the vehicle. Also,as we shall show, it still fails to detect rollovers when rollovers are induced by vertical roadinputs or other external inputs.


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In order to detect tripped rollovers, which happen due to tripping from external inputs, a newrollover index should include the influence of road and other external inputs. Figure 5.3 shows avehicle rollover model that includes the influence of road inputs, and . The figure alsoshows the normal forces on the vehicle, and .

Figure 5.3: Vehicle Model for Un-Tripped and Tripped Rollovers

To derive a rollover index in this case, four-degrees of freedom in the model are needed asdescribed in section 5.4. When the influence of road inputs is included, the normal forces on thevehicle, and , are defined by

2 , (5.3)where

(5.4) and are the right and left dynam 2 and left unsprung masses.

The right and left dynami c vertical tire forces,

ic vertical tire forces, and and , are right

and , can be calculated by

(5.5)(5.6)where is vertical tire stiffness,

,and t and left un

and

are righ sprung mass position

It should be noted that the right and left unsprung m

are right and left unknown road profile inputs,

s.

ass positions, and

, depend on the

dynamic motions of the heave (sprung mass position), ro ll anglethe vertical motion of each

of the vehicle body, andside of the unsprung masses, and . Therefore, the

measurements of roll angle, , and lateral acceleration, , alone are not enough to calculate therollover index for predicting tripped rollover. A lot of additio nal variables need to be measuredor estimated. Moreover, some variables such as unknown road inputs, and , verticaldisplacements of unsprung masses, and sprung mass, cannot be directly measured by sensors.


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Therefore, it is neces sary to develop an approach for estimating the normal forces of a vehicle,and using available sensors only.In the next section, we will present the vehicle dynamic model for un-tripped and trippedrollovers. We will write the model in a form that is suitable for estimating the unknown normal

forces of a vehicle,5.4.

and . Subsequently the rollover index can be calculated.Vehicle Dynamics Model [22]

In order to obtain the rollover index for predicting tripped rollovers, a model of a vehicle with 4-degrees of freedom

is needed as shown in Figure 5.4. The vehicle body is represented by the

sprung m ass while the mass due to the axles and tires are represented by unsprung massesand . The springs and dampers between the sprung and unsprung masses represent the

vehicle suspension. The vertical tire stiffness of each side of the vehicle is represented by thespring .

Figure 5.4: Four-Degrees of Freedom Vehicle Model.

The 4-degrees of freedom of the model are the heave , roll angle of the vehicle body, and thevertical motion of each side of the unsprung masses, and . The variables andthe road profile inputs that excite the system.

The dynamic suspension forces are given by

are

2 2 ,where is suspension stiffness,

(5.7)

is suspension damping.2 2 , , andtheir equilibrium points.Then, the dynamic equations of sprung mass heave and sprung mass roll motions are given by

(5.8)

are measured from

(5.9) and (5.10).


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2 (5.9)

2

2 2 2 2 is mom where ent of inertia, and is distance between right and left suspension. It should be

(5.10)

noted that th e roll dynamics depend o n the lateral dynamics through the lateral acceleration term. By avoiding further expansion of this term in terms of lateral tire forces and lateral dynamic

states, a complicated coupled se t of equations between roll and lateral dynamics is avoided.Instead, the variable is assumed to be measured.

Assume that . Then, the dynam ic models of the right and left unsprung massmotions are given by , 2 2

The equations (5.3) and (5.4) show that

the normal forces on the vehicle,

2 2 and

. (5.11)

(5.12)

com , can be

puted if we know the right and left vertical tire forces, and . In order to design an observer to estimate the variables used to calculate the rollover index, weneed to rewrite equations (5.9)-(5.12) into a state space form. The state space form of equations(5.9)-(5.12) is shown in equation (5.13).

where , (5.13) is the vector of known inputs, is the vector of unknown inputs,

0 1 0 0 0,0 0 00 0 0 1

0 0 0 00 0 0 0 ,0 1

(5.14)

2 2 2 2 0 0

̅


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2 0 2 ,

0 2

, 0 2

0 00 0

2 2 0 00 0

, .The available m 0 0 easurements used to estimate vawhere

,riables is shown in equation (5.15) and (5.17).

(5.15)

, 0 0 0 0 0 0

,

0 0

0,

0 (5.16)

.

where Ψ (5.17)

1 0 0 0 0 0 It is desired to use an observer based on theequations (5.13) and (5.17) show th

, 0 0 0 0 0 1, Ψ . (5.18)above vehicle model to estimate variables. Theat we have the nonlinear terms,

0 and

0Ψ, and the unknowninputs included in the dynam c and measurem problem is zero,

problem are in the

i. We do not have to differentiate

, directly relate to the unknown inputs,appropriate state space form

chapter 8 to solve the problem.

ent equations. Also, the relative degree for thisthe equation (5.15). The measurements,

. Thus, the dynamic and measurement equations of thisfor applying the nonlinear observer described in


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The form of equations (5.13), (5.15), and (5.17) is the same as the form of equations (4.1) -(4.3) .So, we can design a nonlinear observer for this problem.

5.5. Observer Design for the Vehicle Problem

In this sec

,tion, we will apply Theorem 5 and the Corolla

ry to Theorem

3 for the estimation

problem. To apply them, first, we need to calculate the m, atrix . Then, the observability of the pair of needs to be examined. If the pair of is detectable, we can apply Corollary toTheorem 3 for observer gains.Using Theorem 5, the unknown inputs, , can be com

Substitu

te equation (5.19) into (5.13 , puted by

(5.19)

Then, are given by ,).

, , ,

,Φ,(5.20)

(5.21)

However, we found that the pairdesign an observer for this problem

, Φ , is unobservable. W

. We need to modify the equations (5.13) and (5.14).

,, (5.22)ith this system, we cannot directly

The modified model is given by

,

0 1 0 0,

0, 0(5.23)

0 0 (5.24)

0 0 0 1

0 0

0 00 0 0 0 0 1 , (5.25)

2 2 2

2

̅ ̅ ̅

̅


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0 2 2

,

0

2

2 ,

0 2 2 10 0

0 0 0 0 1 ,

where , . 0 00 0 The modi fied m

0odel is valid for designing an 0 0

observer since the original model includes

stabilized terms, and . Now, the modified model will beunobservable only when .

To make the problem easy to solve, the measurement model is given by

2 1 0 0 0 0 0 0 0 0 01 Jacobian of the nonlinear function are

2 where . With this modification, bounds on thedecreased.

0 . (5.26)5.5.1. Observer Design Using Corollary

0 to Theorem 3 in Chapter 4

Apply Corollary to Theorem

Φ

Φ , 3 for

,the problem. The nonlinear function is

0 0 0 0 (5.27)

0

2 2


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For the nonlinear function in equa tions (5.27), the scaling factor z Φ 0, and : ,Φ 0 then 34 and the scaling factor is is 2. Since 6 2 . 6 3. Then, the jacobian of the nonlinear function

,. (Note: we can choose

| puted to find

, such thatΦis com |

and

| and

Φ are sm,|

all.)

2 2 Φ (5.28)

26,5 2 maxΦ, 6,5

2 minΦ Φ Φ(5.29)

(The other elements of

6,6 2 max and are zeros.)

, 6,6 min (5.30)

2The nonlinear function is

(5.31)

ΨΨ 2 For the nonlinear function in equation (5.32), the scaling factor

0 . Ψ (5.32)

the nonlinear function

: ,such that

0 the scaling factor is is 1. Since and. ( 2 6 1 1 1 . |

Then, the jacobian of

| |Ψ is com

| puted to find and

and

. (Note: we can choose

,

are small.)

Ψ 2 Ψ (5.33)

Ψ (5.34)(The other elements of

1,5 1 max , 1,5 1 min Next, we solve equations (4.22) for the observerThe LMI toolbox in Matlab provides only one ga

and are zeros.)

gain using the LMI toolbox in Matlab. (Note:in, though theoretically many solutions can

exist to the LMI (4.22). If a faster convergence rate is desired, the RHS in equation (4.22) could be replaced by a negative definite matrix instead of zero.)

̅ ̅

̅ ̅ ̅ ̅ ̅ ̅ ̅

̅ ̅ ̅ ̅


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5.6. Simulation and Simulation Results

5.6.1. Simulation Setup

In this section, we will evaluate the algorithm described in the previous section in simulations byimplementing it in CARSIM, an industry standard vehicle dynamics simulation software. Thevehicle model from CARSIM chosen for this simulation is a standard SUV.

In the simulation, we simulate the case that the SUV vehicle strikes road bumps duringcornering. The curvature of the road is shown on Figure 5.5 and the road bumps are shown onthe second row of Figure 5.6. The first bump is applied to the right wheels of the vehicle and thesecond bump is applied to the left wheels of the vehicle. The magnitude of the first bump islarger than that of the second bump but the displacement rate of the first bump is slower than thatof the second bump. The fast displacement rate of the road input causes a lot of change in normaltire forces. The vehicle is set up to make cornering with a vehicle speed of 100 kph. It should benoted that when the vehicle strikes a bump, the wheels of the vehicle still do not lift off the roadsurface. The lateral acceleration in this case is shown on the first row of Figure 5.6.

Figure 5.5: Road Curvature

Figure 5.6: Lateral Acceleration and Road Inputs

0 5 10 15 20 25-0.01

-0.005

0

0.005

0.01Road Curvature

time (sec)

C u r v a

t u r e o

f p a

t h ( 1 / m )

0 5 10 15 20 25-10

-5

0

5

10Lateral Acceleration Input

time (sec)

a y

( m / s e c 2

)

4 6 8

0

0.1

0.20.3

0.4

Road Inputs, Z rr

time (sec)

R o a

d I n p u t s ,

Z r r ( m )

14 16 18

0

0.1

0.20.3

0.4

Road Inputs, Z rl

time (sec)

R o a

d I n p u t s ,

Z r l ( m )


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5.6.2. Simulation Results

For designing an observer for this problem, we assume that

1.

45 45 .,

2.

1 1 1rad./sec. or

5 deg./sec.,

3. 15 m/s2,

4. 57.3 57.3 is small.The feedback measurements that we need to feed to the observer are . Right unsprung massvertical acceleration, , 2. Left unsprung mass vertical acceleration, , 3. Right suspensioncompression,

Then the local bound of the nonlinearity can be found. The observer gain f

1 , and 4. roll rate, . observer is

The estimation results are shown on Figure 5.7-F 13.57621.4320 109.8388states are very close to the actua440.6767 0.8108 432.667220.0510 ound by using the new

5.288 .

l values. The estim13.5380igure 5.10. The results show that the estimatedated states are not exactly equa8.1034l to the actua113.1lvalues because the vehicle model in CARSIM has many degrees of freedom but our model isonly a 3-degrees of freedom model with 6 states. Also, there is a time delay in CARSIM betweenthe front and rear wheels of the vehicle when the vehicle strikes a bump.

Figure 5.7: Estimation of Right and Left Suspension Compressions

0 5 10 15 20 25

-0.2

-0.1

0

0.1

zs -zur : solid: ac tual, dotted circ le : estimate

time (sec)

z s - z

u r

( m )

0 5 10 15 20 25

-0.1

0

0.1

0.2

zs -zul: solid: ac tual, dotted circ le : estimate

time (sec)

z s - z

u l

( m )

Note: the ou tputs of

and from CARSIM and compare th

the Carsim are right and left suspension compressions,

. However, on Figure 5.7, we compute the actual value of ande values of them with those from our observer.


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Figure 5.8: Estimation of Right and Left Suspension Compression Rate

0 5 10 15 20 25

-0.5

0

0.5

d(z s -dzur )/dt: solid: ac tual, dotted circ le : estimate

time (sec)

d ( z s

- d z u

r ) / d t ( m / s e c )

0 5 10 15 20 25

-0.5

0

0.5

d(z s -dzul)/dt: solid: ac tual, dotted circ le : estimate

time (sec)

d ( z

s - d z u

l ) / d t ( m / s e c )

Note: the outputs of the Carsim are right and left suspension compression rates,and . However, on Figure 5.8, we compute the actual values ofand and compare these values with our observer.

Figure 5.9: Roll Angle and Roll Rate Estimation.

0 5 10 15 20 25

-0.2

-0.1

0

0.1

: solid: ac tual, dotted circ le : estimate

time (sec)

(

r a d )

0 5 10 15 20 25

-0.5

0

0.5

d /dt: solid: actual, dotted circle : estimate

time (sec)

d / d t ( r a

d / s e c

)


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Figure 5.10: Normal Tire Forces Estimation, F zr and F z

0 5 10 15 20 250

5000

10000

15000Fzr: solid: ac tual, dotted circ le : estimate

time (sec)

F z r

( N )

0 5 10 15 20 250

5000

10000

15000Fzl: solid: actual, dotted circle : estimate

time (sec)

F z l

( N )

The estimated rollover index is shown on Figure 5.11. The result shows that the estimated andactual rollover indices are extremely close. There are very small errors and these happen

because there is a time delay between the front and rear wheels of the vehicle when the vehiclestrikes a bump. Since we use only 3-degrees of freedom for observer design, we cannot providethis time delay. However, if we carefully look at the result, we will see that the estimatedrollover is good enough to predict un-tripped and tripped rollovers.

Figure 5.11: Rollover Index Estimation

0 5 10 15 20 25-1

-0.5

0

0.5

1Rollover Index: solid: actual, dotted circle : estimate

time (sec)

R o l

l o v e r

I n d e x

5.7. The Scaled Vehicle for Experiments

The simulation results in the previous section show that the new rollover index can detect bothtripped and un-tripped rollovers. However, we still need to confirm the approach with a testvehicle.

The use of a full-sized vehicle for testing of a control system in this rollover application ischallenging due to cost limitations and safety issues. It is estimated that the cost of developing afull-sized instrumented vehicle for testing is more than $100,000 [23]. Most of this cost goes forequipment and instrumentation development. Also, a full-sized vehicle is difficult to operate andinvolves significant safety issues in this application which involves vehicle rollover.


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It is more convenient to use a scaled vehicle to test the roll control system. A scaled radiocontrolled vehicle is inexpensive and safe for evaluation of rollover maneuvers. Manyresearchers have studied scaled vehicles for testing of vehicle dynamics and control systems. Forinstance, references [23], [24], [25], and [26] developed a 1/10 th scaled vehicle to study lateralvehicle dynamics. Reference [27] developed a 1/8 th scaled vehicle to study longitudinal vehicle

dynamics. References [28], [29], [30], and [31] studied stability control algorithms with a scaledvehicle. Reference [32] studied tire characteristics with scaled tries.

In order to use a scaled vehicle to describe the behavior of a full-sized vehicle, we need to showthat they have dynamic similarity. In the next section, we will show that the scaled vehicle weuse has dynamic similarity to a full-sized vehicle. Then, we describe the experimental setup totest rollover scenarios and present the experimental results in the following section.

5.7.1. Dynamic Similitude Analysis

Two systems of different size scales are dynamically similar if the solutions to their governingdifferential equationthe equations of mBuckingham

s are identical after accounting for the dimensional scaling of parameters inotion. There are many approaches to evaluate dynamic similarity. The

theorem is a tool that provides us an easy approach to show dynamic similarity.The details of the Buckingham theorem are discussed in [27], and [33].

This theorem is convenient to apply. It can show dynamic similarity without explicitly knowingthe accurate dynamic equations of b oth systems. We need only know the list of all variables and

parameters associated with the system. References [23], [24], [25], and [32] used theBuckingham theorem to show the lateral dynamic similarity between a full-sized vehicle and a1/10 th scaled vehicle. Also, reference [27] showed the longitudinal dynamic similarity between afull-sized vehicle and a 1/13 th scaled vehicle. Therefore, we use the Buckingham theorem toshow the roll and vertical vehicle dynam ics similarity between our scaled vehicle and a full-sizedvehicle.

The Buckingham theorem shows that if the values of the dimensionless groups of variablesand parameters are m aintained, then the solution to any differential equation, regardless of itsorder or nonlinearity, can be made invariant with respect to dimensional scaling [26]. To applythe Buckingham theorem to the roll and vertical vehicle dynamics, we need the list of allvariables and parameters associated with these dynamics. We limit the number of variables and

parameters to those used in the 4-degrees

, , , ,

of freedom

,

model as described in section 5.4.Therefore, the roll angle will be a function prim

,arily d

, ,epende

, ,nt on the s

, .caled parameters,

The param (5.35)

eters and variables of the vehicle dynamic model in equation (5.35) and their units aregiven in Table 5.1.


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Table 5.1: Summary of Parameters Associated with the Vehicle Dynamics

Mass Length Time, roll angle dimensionless 0 0 0

ssa, sprung m kg 1 0 0ssa, unsprung m kg 1 0 0

, moment of inertia kg.m 2 1 2 0, spring stiffness kg/sec 2 1 0 -2

er p, dam kg/sec 1 0 -1, distance between left and right

suspensions m 0 1 0

, c.g. height m 0 1 0, tire stiffness kg/sec 2 1 0 -2

, lateral acceleration 2m/sec 0 1 -2, sprung mass vertical acceleration 2m/sec 0 1 -2

, left unsprung mass vertical acceleration 2m/sec 0 1 -2

, right unsprung mass verticalacceleration

2m/sec 0 1 -2

There are 13 parameters ( to represent the vehicle dynamics and 3 basic unit dimensions

(j=3): mass (M), length (L), and time (T). W e choose the sprung mass ( ), c.g. height ( ), andlateral acceleration ( ) to represent repeating fundamental units in three-dimensional space. So,the rem aining 10 unused parameters (k=n-j=10) can be formed as dimensionless groups byappropriate division or multiplication of the repeating variablesthe

, , and . The lis t of allgroups is given in Table 5.2.

13


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Table 5.2: π Groups

[M]0[L]0[LT -2]0[] , ,, [M]1[L]0[LT -2]0[M] -1 ,, ,

[M]-1

[L]-2

[LT-2

]0

[ML2

]1

,, , [M] -1[L]1[LT -2]-1[MT -2]1 ,, , [M] -2[L]1[LT -2]-1[MT -1]2 ,, , [M]0[L] -1[LT -2]0[L]1 ,, , [M] -1[L]1[LT -2]-1[MT -2]1 ,, ,

, [M]0[L]0[LT -2]-1[LT -2]1

, , , [M]0[L]0[LT -2]-1[LT -2]1 , , ,,, [M]0[L]0[LT -2]-1[LT -2]1 Note: All of the dimensionless parameters, such as angles form their own group.

To have the roll and vertical vehicle dynamics of the scaled vehicle the same as those of a full-sized vehicle, we need to tune the sc aled vehicle until the values of groups of the scaledvehicle are close to the values of groups of a full-sized vehicle. The variables and parametersof the full-sized and scaled vehicle are shown in Table 5.3 and the groups of them are shown inTable 5.4. The photographs of the scaled vehicle are shown in Figure 5.12.

Figure 5.12: Scaled Test Vehicle: 1:8 (30.5 x 58.5 cm)


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Table 5.3: Vehicle Variables and Parameters

Variables and Full-SizedScaled VehicleParam eter vehicle (kg) 3 1600 (kg) 0.2 135

(kg.m 2) 0.04 600(N/m) 900 90000

(N.sec/m ) 15 3000

(m) 0.2 1.11 (m) 0.18 1

(N/m) 4000 400000

Note: The parameters of the full-sized vehicle are obtained from the software CARSIM.

Table 5.4: Comparison of π Groups

Scaled Vehicle Full-Sized v ehicle 15 11.9 0.41 0.375

4.5541.11 1.11⁄⁄ 56.25⁄ ⁄ 3.5

⁄240 250 ⁄ ⁄⁄

⁄⁄

⁄ ⁄Table 5.4 shows that if the lateral accelerations ( and ) that cause the scaled and full-sizedvehicle to roll over have the same magnitude, then the group values of the scaled vehicle are

close to those of the full-sized vehicle. Also, the resulting vertical accelerations will have thesame order.

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