Evaluation of a Pulsed Active

Steering Control System

R. Vos

DCT 2009.010

Traineeship report

Coach: Prof. J. McPheeSupervisor: Prof.dr. H. Nijmeijer

Technische Universiteit EindhovenDepartment Mechanical EngineeringDynamics and Control Group

Eindhoven, February, 2009

Acknowledgements

I would like to thank my supervisors, Prof. J. McPhee and Prof. A. Khajepour for the op-portunity to do this internship at the University of Waterloo and for their support, guidanceand knowledge.

I would also like to thank A. Abdel-Rahman for all his help and support throughout theinternship and for the contribution he made to my work.

Finally, I would like to thank my family and all my friends for their support and encourage-ment to make this achievement possible.

i

Abstract

In this report the effect of a Pulsed Active Steering Control system (PASC) on a vehicletrajectory and rollover is studied. Former studies have shown that this system is able toprevent rollover better than the Active Steering Control system, the Direct Yaw MomentControl system and the Integrated Control system. However, different pulse forms, frequenciesand amplitudes show different effects on the vehicle trajectory and rollover. These effects areinvestigated in more detail in this report by simulating J-turn maneuvers using a standardvehicle with the software program ADAMS. The vehicle trajectory is directly given by theprogram, whereas the vehicle rollover is investigated by studying the rollover coefficient. Theprimary goal of the PASS is to decrease the vehicle rollover and therefore, simulations areperformed using a steering wheel input with a subtracted pulse. The secondary goal is to usethe system for track following and therefore, the pulse is added to the steering wheel input.

The simulation results show that both the amplitude and the frequency of the pulse havea big effect on the vehicle trajectory and rollover coefficient. A high frequency reduces therollover coefficient the most and gives the best combination of vehicle trajectory and rollover.The amplitude of the pulse can be altered to find a specific vehicle trajectory and to reducethe rollover coefficient below a certain threshold. A C1 continuous non-symmetric pulse isable to reduce the rollover coefficient the most compared to a symmetric pulse and a C0

continuous non-symmetric pulse.The results found with ADAMS are validated by comparing different simulation results

obtained by ADAMS with simulation results obtained by the software program Maple andDynaFlexPro. The programs show different results due to the difference in the models used,but these results are consistent for different pulse forms and frequencies.

A pulsed actuation system is designed to be built in a test setup. The system consists ofa gear-train assembly and a pulse actuator. The gear-train assembly comprises 4 spur gearsand a planetary gear-set. A worm-gear is taken as pulse actuator. All the gears of the systemare chosen such that a pulse with a maximum frequency can be applied to the steering wheelcolumn and such that they can handle the torque and power supplied by an available motor.

ii

List of Figures

3.1 Vehicle motions defined according to the SAE convention . . . . . . . . . . . . . . . . 83.2 Representation of the J-turn maneuver input . . . . . . . . . . . . . . . . . . . . . . 103.3 Representation of the used symmetric and non-symmetric pulse . . . . . . . . . . . . . 103.4 Nonlinear Vehicle Yaw Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.5 Nonlinear Vehicle Roll Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.6 The pulsed and un-pulsed steering wheel input s for the J-turn maneuver . . . . . . . . 123.7 Vehicle trajectory for a pulse with an amplitude of 120 and 80 degrees for different frequencies 133.8 Rollover coefficients for a pulse with an amplitude of 120 and 80 degrees for different frequencies 143.9 Vehicle trajectories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.10 Rollover coefficient for the non-symmetric pulse with an amplitude of 120 degrees . . . . . 163.11 Vehicle trajectory and rollover coefficients for different pulse forms . . . . . . . . . . . . 173.12 Representation of the different pulse forms . . . . . . . . . . . . . . . . . . . . . . . 183.13 Rollover coefficient for inputs with different subtracted pulses and for an input with a constant

subtracted value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.14 The un-pulsed and pulsed steering wheel angle input and vehicle trajectory . . . . . . . . 203.15 Rollover coefficient for different pulse forms and for the constant added value . . . . . . . 20

4.1 Self-aligning moment in ADAMS and Maple for a symmetric pulse with a frequency of 1 Hzand 4 Hz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4.2 Self-aligning moment in ADAMS and Maple for a non-symmetric pulse with a frequency of1 Hz and 4 Hz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

5.1 Gear-train assembly design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285.2 multiple-bar mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305.3 adjustable-amplitude mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315.4 Steering system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335.5 Torque versus steering wheel acceleration . . . . . . . . . . . . . . . . . . . . . . . . 345.6 Torque and power versus the frequency for different pulse forms and peak-time values . . . 355.7 Torque and power versus the amplitude for the symmetric pulse for different frequencies . . 355.8 Torque and power versus the amplitude for the C0 continuous non-symmetric pulse for dif-

ferent frequencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365.9 Torque and power versus the amplitude for the C1 continuous non-symmetric pulse for dif-

ferent frequencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

A.1 Rollover coefficient for a pulse with a frequency of 8 Hz and an amplitude of 120 degreesbetween t = 1.5 - 5.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

C.1 Torque (Nm) versus speed (rpm) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

iii

D.1 Tooth parts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

iv

List of Tables

3.1 Parameters of the used demo vehicle model . . . . . . . . . . . . . . . . . . . 8

5.1 maximum frequency for each ratio based on the maximum rotational speedand torque . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

5.2 Planetary gear-set and worm gear data . . . . . . . . . . . . . . . . . . . . . . 38

B.1 Values a, b and c for different frequencies used for the non-symmetric pulse . 44

C.1 Technical data of the motor . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

v

Glossary

NHTSA National Highway Traffic Safety AdministrationSUV Sports Utility VehicleASC Active Steering ControlDYC Direct Yaw Moment ControlHC Hybrid ControlARS Active Rear wheel SteeringPASC Pulsed Active Steering Controls steering wheel angle steering angle of the front wheelsis steer ratioL wheelbaseR corner radiusay lateral acceleration understeer coefficientg gravity constantA pulse amplitudef pulse frequencyt timea pulse peak-time valueb falling slope valuec rising slope valueR0 rollover coefficientFz,R vertical tire load on the vehicles right hand sideFz,L vertical tire load on the vehicles left hand sidems vehicle sprung massM total vehicle massTr track widthh height of CG above grounde distance between CG and roll axisay,s lateral acceleration of sprung massvy lateral acceleration of total massu longitudinal velocityr yaw rate

vi

roll acceleration of vehicleT vibration timeMz self-aligning momentRi radius of gear ii rotational speed of gear iz ratio between ring-gear and sun-gearTi torque on gear iN number of teeth on gearP diametral pitch of gearD pitch diameter of gearTs torque on steering wheelIeq equivalent inertia of steering systembeq equivalent damping of steering systemkeq equivalent stiffness of steering system

s acceleration of steering wheel

s angular velocity of steering wheels angle of steering wheelMz self-aligning momentr scale factorTps torque delivered by power steering systemPs power on steering wheelR ratio between worm-gear and ring-gearsun angle of sun-gearA pulse amplitudef pulse frequencyt timesun rotational speed of sun-gearsun,max maximum rotational speed of sun-gearTsun torque on sun-gearTsun,max maximum torque on sun-gear

vii

Contents

1 Introduction 1

1.1 Research goals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Report Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Literature Review 3

3 Pulsed Active Steering effects 7

3.1 ADAMS simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.2 Vehicle dynamics with pulse subtraction . . . . . . . . . . . . . . . . . . . . . 12

3.2.1 Symmetric pulse input . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.2.2 Non-symmetric pulse input . . . . . . . . . . . . . . . . . . . . . . . . 153.2.3 Optimal subtraction method . . . . . . . . . . . . . . . . . . . . . . . 17

3.3 Vehicle dynamics with pulse addition . . . . . . . . . . . . . . . . . . . . . . . 193.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

4 Results validation 23

4.1 DFP and Maple simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.2 Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

5 Pulse actuation system 27

5.1 Gear-train assembly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275.2 Pulse actuator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305.3 Power/Torque calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325.4 Worm-gear design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

6 Conclusions and recommendations 40

6.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 406.2 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

Bibliography 41

A Pulse during an extended time 43

B Non-symmetric pulse values 44

C Motor characteristics 45

viii

D Calculation ring gear thickness 46

ix

Chapter 1

Introduction

Statistics from the National Highway Traffic Safety Administration (NHTSA) show that 9,362of the total of 30,521 traffic fatalities in the United States in 2006 are due to rollover of thevehicle. Sport Utility Vehicles (SUVs) had the highest rollover involvement rate of any vehicletype in fatal crashes: 35 % for SUVs, 28% for pickups, 17 % for vans and 17 % for passengercars. In 1996 8,318 fatalities occurred due to rollover of the vehicle, so the amount of rollovercrashes is increasing [1]. To decrease the amount of accidents due to rollover of the vehicle,a strategy needs to be designed to control the vehicle (dynamics) to improve the safety andride comfort of the vehicle.

Much research has already been performed in the vehicle motion control area to control thevehicle (dynamics). Four main control techniques have been studied widely. One techniquefocusses on controlling the steering angle of the front wheels (Active Steering Control, ASC),one focusses on controlling the braking force distribution on all the four wheels (Direct YawMoment Control, DYC), one focusses on controlling both the front wheels and the brakingforce distribution (Hybrid Control, HC) and one technique focusses on controlling the steeringangle of the rear wheels (Active Rear Wheel Steering, ARS).

1.1 Research goals

Kuo [8] has investigated if the ASC system, the DYC system and the HC system are ableto prevent vehicle rollover. He claims to have shown that none of these systems are efficientenough to decrease the rollover of the vehicle. Therefore, he has proposed the Pulsed ActiveSteering Control System. He states that this new system is able to lower the chance of vehiclerollover efficiently. However, different pulse forms, frequencies and amplitudes show differenteffects on the vehicle rollover and on the vehicle trajectory. Since rollover crashes can occur forexample by avoiding an obstacle on the road, it is also important that the vehicle trajectoryis not changed too much due to the anti-rollover system. The exact effects of the PulsedActive Steering Control system on both the rollover as the vehicle trajectory have not beenstudied into detail by Kuo, so more investigation needs to be performed. To study mechanicaleffects of this Pulsed Active Steering System on the total steering system and to validate thesimulation results experimentally, a test setup needs to be build as well.

1

The goals of this project, based on the work done by Kuo, are therefore:

Investigate the effect of the PASC system on the vehicle rollover and trajectory

Design and optimize a pulse actuation system for a test setup.

1.2 Report Overview

The overview of this report is as follows:Chapter 2 presents information about the four main control techniques used to control the

vehicle dynamics and will describe if these systems are able to control the vehicle trajectoryand rollover for different driving maneuvers and circumstances.

Chapter 3 gives information about the simulations performed to investigate the effect ofdifferent pulse forms, amplitudes and frequencies on the vehicle trajectory and rollover andshows the simulation results. The effects are investigated for a steering wheel input with asubtracted pulse and with an added pulse. The effects are compared to a steering wheel inputwith a constant subtracted or added value to see if the Pulsed Active Steering System is ableto reduce the vehicle rollover more than the Active Steering System.

Chapter 4 describes the validation of the simulation results described in Chapter 3. Thisis done by comparing the self-aligning moment obtained by the software program ADAMSwith the self-aligning moment obtained by the software program Maple and DynaFlexPro.

Chapter 5 shows the proposed pulse actuation system, consisting of a gear-train assemblyand a pulse actuator. Upon given constraints the gears of the gear-train assembly are chosen.A worm-gear is chosen as pulse actuator and is further designed to be able to apply anoptimized maximum frequency to the steering wheel input. For this the maximum torqueand power needed on the steering wheel column are calculated using an analytical model.

Chapter 6 presents the conclusions made upon the simulations performed in this reportand discusses some possible future research to improve the Pulsed Active Steering System.

2

Chapter 2

Literature Review

This section will give information about the four main control techniques used to control thevehicle dynamics and will describe if these systems are able to control the vehicle trajectoryand rollover for different driving maneuvers and circumstances. Different controllers havebeen designed for the same control technique and some are therefore also addressed in thissection.

Advantages of active steering for vehicle dynamics control

Ackermann et al. [2] have discussed the potential of DYC and ASC for yaw disturbanceattenuation in terms of physical limits. They state that ASC only requires one fourth of thefront wheel tire force compared to DYC. They also claim that ASC is better to generate acorrective torque to compensate torques caused by asymmetric braking and still have brakingforce left for acceleration. Asymmetric braking can arise due to a so called -split brakingsituation; the contact surface for wheels on the right hand side of the vehicle is dry, while thecontact surface for wheels on the left hand side of the wheels is icy. If the DYC system gener-ates a corrective torque, no braking force for deceleration is available anymore. Furthermore,the ASC gives more driving comfort and higher safety.

Two vehicle dynamics control concepts have been summarized. The first concept focusseson the control of the yaw motion and consists of decoupling of the vehicles yaw and lateralmotion as first presented in [3]. They state that this concept separates two basic taskswhich have been the responsibility of the driver up until now: path following and disturbanceattenuation. The first task is still left to the driver, but the disturbance attenuation can becontrolled by the active steering system, making driving a vehicle easier and safer. Simulationshave shown excellent disturbance rejection in -split braking and side-wind maneuvers.

The second concept focusses on vehicle rollover avoidance by active steering and brakingas first proposed in [4]. The presented controller consists of 3 feedback loops: emergencysteering control, emergency braking control and continuous operation steering control. If therollover coefficient (defined in [5]) reaches e.g. a value of 0.9 (rollover limit: |R| = 1) dueto a high drivers steering wheel input, the emergency steering control comes into action,the front wheel steering angle is reduced and rollover of the vehicle is avoided. At the sametime vehicle deceleration occurs through braking and the chance of vehicle rollover is furtherreduced. By controlling the braking pressure the vehicle trajectory is maintained according tothe drivers steering command. The continuous operation steering control is added to improvethe vehicles roll-damping and roll-disturbance attenuation. Simulations have shown that this

3

control setup is able to prevent rollover and is able to maintain practically the same vehicletrajectory as an uncontrolled vehicle.

Study on integrated control of active front steer angle and direct yaw moment

Nagai et al. [6] have proposed a HCS. By using a model-matching control technique, thesystem is designed such that the the performance of the actual vehicle model follows that ofan ideal vehicle model. The actual vehicle model is described as a bicycle model includingdirect yaw moment input. The desired vehicle model has been derived by the control law ofARS in which the rear wheels are steered such that the vehicle body sideslip is zero. Theproposed model-matching controller consists of the desired model and a feed-forward andfeedback compensator. The feed-forward compensator decides the control inputs; the frontwheel steering angle and the direct yaw moment generated from braking forces. The feedbackcompensator is designed to suppress the vehicle body sideslip angle and the yaw rate response.

Simulating different driving events show that the yaw motion and the sideslip motion ofthe vehicle is improved by this system compared to these motions when only a DYC systemis used. The simulations also show that the system has a robust performance to make theactual vehicle response follow the desired vehicle response.

Evaluation of an Active Steering System

Orozco has investigated the stability and robustness of an ASC system (see [7] and referencestherein) and evaluated this system by simulating different driving events. The inputs of thevehicle model are the steering angle set by the driver and a side wind force. A steering anglecontribution is derived using the yaw rate and the steering wheel angle and this contributionis added to the drivers command. For controller analysis the linear single-track model is used,whereas for the simulations a non-linear two-track model is used.

Different simulations show that a wind force disturbance is reduced by the control system,that the control system is able to react almost twice as fast as a human driver to wind forcedisturbances,that the controlled vehicle is harder to make unstable than the uncontrolledvehicle and that the system is robust and stable.

Sports Utility Vehicle Rollover Control with Pulsed Active Steering Control

Strategy

Kuo [8] has investigated if the ASC system, the DYC system and the HCS are able to preventvehicle rollover. A nonlinear 4 degree of freedom vehicle yaw/roll model as well as a complexnonlinear tire model have been derived and used for these simulations. He claims that thisnew model represents the real-world vehicle to a good degree of accuracy. Using an ASCsystem, the results show that the rollover coefficient is reduced to a small proportion of theoriginal magnitude, but the system is not able to fully reduce the vehicle rollover below acertain threshold when the rollover is too high due to an extreme drivers steering input. TheDYC system also seems to be unable to prevent rollover at high vehicle speeds and extremedriver steering inputs. This is because the high braking forces needed to decrease the rolloverresult in a significant shift in vertical tire load to one of the front tires, causing abnormal tirelateral forces. This results in vehicle instability. The HCS shows better results compared tothe other two controllers, but since it also includes the differential braking mechanism, it issensitive to vertical tire load shift and would therefore also fail to prevent rollover.

4

Therefore, Kuo has designed and tested a slightly new vehicle rollover control strategy, thePulsed Active Steering Control (PASC) system. The difference between the Active SteeringControl system and the Pulsed Active Steering Control system is that, instead of a constantvalue, a pulse with a certain amplitude and frequency is added or subtracted to the steeringwheel input given by the driver. The only input of the designed controller is the steeringwheel input given by the driver. By calculating different variables the rollover coefficientis calculated. If this exceeds a designated threshold a pulse is subtracted from the originaldriver steering input. Simulations show that using a symmetric pulse results in a rolloverwith sudden bumps higher than the rollover obtained for the un-controlled vehicle. Using anon-symmetric pulse results in a vehicle rollover lower than for the un-controlled vehicle is.Therefore, the non-symmetric pulse can best be used to decrease the vehicle rollover. Thenon-symmetric pulse used consists of a smooth curve with a sharp, gradually decreasing slopecombined with a smooth, gradually increasing slope. Compared to a symmetric pulse anda square pulse, this pulse shows a smaller reduction of the rollover coefficient in its totalamount, but it is able to eliminate a sudden bump experienced by using the other two pulses.

Results from several driving maneuver simulations show that this new controller is ableto prevent rollover. However, it is also visible that the controlled vehicle trajectory is dif-ferent from the uncontrolled trajectory. Simulating at different frequencies shows that if thefrequency is either too high or too low, the efficiency of the controller is reduced. It is alsovisible that different pulse frequencies result in different vehicle trajectories. Overall, the sim-ulations show that the pulse amplitude, the pulse frequency and the threshold of the rollovercoefficient to trigger the controller are the three important control variables essential for awell-designed PASC system.

Improving Yaw Dynamics by Feed-forward Rear Wheel Steering

Besselink et al. [9] have discussed two control systems for ARS to improve the vehicle yawdynamics. The results of these controllers have been compared with a simulation modelbased on an enhanced bicycle model. In this model the tire relaxation length and suspensionsteering compliance have been taken into account. The first controller, the yaw rate feedbackcontroller, consists of a reference model and a rear wheel steering controller. The controlleris designed to minimize the yaw rate overshoot, since this overshoot is undesirable and leadsto an increased workload for the driver. The reference model provides the reference yaw rateand is compared to the actual vehicle yaw rate. The difference is fed back to the steeringcontroller. Simulations show that this active rear wheel steering control system is able tosuppress the undesired yaw rate overshoot.

The disadvantage of this controller is that on a real vehicle an accurate yaw rate signalis needed, but the yaw rate signal given by an ESP sensor does not meet the requirements.Simulations also show that the required rear wheel steering angle needed to eliminate theyaw velocity oscillation is not related to the frequency of the original yaw oscillation. Thismeans that it is not necessary to apply counter steering at the rear wheels depending on theyaw velocity oscillation. Therefore, a feed-forward rear wheel steering controller has beendesigned. Using the relation between the step response of the rear wheel steering angle andthe front steering angle, as found for the feedback controller, a transfer function is proposedto relate the steering angle of the rear wheels to the steering angle of the front wheels.For this controller only the front wheel steering angle and the vehicle forward velocity arenecessary. Simulations show that this controller is able to eliminate the yaw velocity overshoot

5

and oscillations without the need of an accurate yaw rate sensor. The performances of thissystem are almost the same as for the feedback controller.

6

Chapter 3

Pulsed Active Steering effects

To investigate the effect of a Pulsed Active Steering Control system (PASC) on the vehicletrajectory and rollover, simulations are performed using a steering wheel input with differentsubtracted or added pulse forms, frequencies and amplitudes. The simulations are performedwith the mechanical system simulation software program MSC.ADAMS.

The first goal of the PASC is to decrease the vehicle rollover as much as possible withoutchanging the vehicle trajectory too much. This can be done by decreasing the drivers steeringinput. Therefore, the effects of a steering wheel input with a subtracted pulse is investigatedfirst. The second goal of the PASC is to use the system for track following. The deviationfrom a desired trajectory due to understeer for example can be decreased by increasing thedrivers steering input. Therefore, the effects of a steering wheel input with an added pulseis investigated second. Some of the results are compared to a steering wheel input with aconstant (un-pulsed) subtracted or added value. This is done to investigate if the PASCsystem works better than the ASC.

3.1 ADAMS simulations

The software program MSC.ADAMS makes it possible to simulate the full-motion behaviorof a complex mechanical system and to analyze multiple design variations or motion inputsin a fast way. All the simulations are made using the non-linear demo vehicle model providedby the program. The tire-model used is Pacejka 2002 consisting of the Magic Formula forboth longitudinal and lateral tire forces, the transient response to friction changes and theslip dependent relaxation effect. Parameters of the demo vehicle model are shown in Table3.1. The vehicle motions are defined according to the SAE sign convention, as indicated inFigure 3.1.

The steering wheel angle (s) given by the driver is chosen as input for all simulations.The resulting steering angle of the front wheels () can be found by dividing the steeringwheel angle by the steer ratio (is). The steer ratio of the vehicle model used can be found bysimulating steady-state cornering. For steady-state cornering the steering angle of the frontwheels can be found by the equation:

=L

R+ay

g (3.1)

7

Definition Symbol Unit Value

Total vehicle mass m kg 1530Vehicle sprung mass ms kg 1430Wheel base L m 2.56Track width front wf m 1.52Track width rear wr m 1.59Distance from center of gravity to front axle Lf m 1.48Distance from center of gravity to rear axle Lm m 1.077Height of center of gravity above ground h m 0.432Spring stiffness K N/m 1.25e5Vehicle moment of inertia w.r.t. x-axis Ixx kgm

2 584Vehicle moment of inertia w.r.t. y-axis Iyy kgm

2 6129Vehicle moment of inertia w.r.t. z-axis Izz kgm

2 6022

Table 3.1: Parameters of the used demo vehicle model

Figure 3.1: Vehicle motions defined according to the SAE convention

8

With L the wheelbase, R the corner radius, ay the lateral acceleration and the understeercoefficient of the vehicle and g the gravity constant. The understeer coefficient determineswhether the steering angle needs to be changed to remain a certain constant radius R if theforward speed of the vehicle is increased. For a neutral vehicle the understeer coefficient iszero and the steering angle can remain the same, for a understeered vehicle the understeercoefficient is higher than zero and the steering angle needs to be increased and for an over-steered vehicle the understeer coefficient is lower than zero and the steering angle needs tobe decreased. Simulating steady-state cornering at different vehicle speeds shows that thevehicle model used in ADAMS has understeer. The exact understeer coefficient has not beendetermined, since it is not important for the investigation performed in this report. To calcu-late the steer ratio the steady-state cornering needs to be simulated at a low vehicle speed. Atlow speeds the lateral acceleration of the vehicle is very low and the effect of the understeercoefficient can therefore be neglected. The equation of the steer ratio than becomes:

is =s

=

sLR+

ayg=

sR

L(3.2)

Using a steering wheel input of 300 degrees for the steady-state cornering simulation a result-ing corner radius of 11.5 meters is found. These values result in a steer ratio of 23.5 for thedemo vehicle model.

The driving maneuver and the different pulse forms used for the simulations and a wayto investigate the vehicle rollover are described next.

Driving maneuver

It is expected that the influence of different pulse forms, frequencies and amplitudes on thevehicle trajectory and rollover is higher when the vehicle is rolling or skidding. Therefore,a relatively extreme driving maneuver, the J-turn maneuver, is chosen to be simulated. Arepresentation of the simulation input for this maneuver can be found in Figure 3.6. As canbe seen, after one second the steering input gradually increases to a maximum within onesecond and stays here from the 2nd to the 5th second. From the 5th to the 6th second thesteering input gradually decreases back to 0 degrees.

Pulse forms

The effect of two different pulse forms are investigated: a symmetric pulse and a non-symmetric pulse. The symmetric pulse is given by the following equation:

y(t, f) = A(1 cos(2pift)) (3.3)

with A the pulse amplitude, f the pulse frequency and t the time. A representation of thissymmetric pulse is shown in Figure 3.3. The non-symmetric pulse is the one recommendedby Kuo. According to his findings, this special pulse form is more useful in reducing therollover coefficient than the symmetric pulse is. The pulse form consist of a sharp, graduallydecreasing slope (given by y1) combined with a smooth, gradually increasing slope (given byy2):

y1(t) = 2Ae(ta)2

b for 0 t a (3.4)

y2(t) = 2Ae(ta)2

c for a t (3.5)

9

0 2 4 6 8 100

0.2

0.4

0.6

0.8

1

time [s]

stee

ring

whee

l ang

le

s [de

g]

Steering wheel input vs time

Jturn maneuver input

Figure 3.2: Representation of the J-turn maneuver input

With A the amplitude of the pulse and t the time. The value a represents the time wherethe pulse reaches its peak value and the values b and c give the shape of the falling and risingslope, respectively. A representation of the shape of this non-symmetric pulse with a = 0.25,b = 0.005 and c = 0.045 is also shown in Figure 3.3.

0 0.2 0.4 0.6 0.8 11

0.8

0.6

0.4

0.2

0

Time [s]

y

symmetric pulsenonsymmetric pulse

Amplitude

b c

a

Figure 3.3: Representation of the used symmetric and non-symmetric pulse

To determine the effects of different frequencies on the vehicle trajectory and rollover, allthe investigations are performed for pulses with a frequency of 1, 2, 4 and 8 Hz.

Vehicle rollover

The effects of the PASC system on the vehicle trajectory can be given directly by the softwareprogram. The effects on the vehicle rollover is investigated by calculating the rollover coef-ficient, which is a measure for the rollover risk [8]. The coefficient is given by the followingequation:

Ro =Fz,R Fz,LFz,R + Fz,L

=2msMTr

{((h e) + e cos)ay,s

g+ e sin} (3.6)

With Fz,R and Fz,L the vertical tire load on the right hand side and the left hand siderespectively, ms the vehicle sprung mass, M the total vehicle mass, T the track width, h the

10

height of the center of gravity above the ground, e the distance between the center of gravityand the roll axis, the roll angle and g the gravity constant. ay,s is the lateral accelerationof the sprung mass and is given by the equation:

ay,s = vy + ur e (3.7)

with vy the lateral acceleration of the total mass, u the longitudinal velocity, r the yaw rateand the roll acceleration of the vehicle. The non-linear vehicle model and all the parametersand variables used are shown in figures 3.4 and 3.5. The vehicle is about to rollover if thetire loads on one side of the vehicle become zero. At that moment the absolute value of therollover coefficient equals 1.

Figure 3.4: Nonlinear Vehicle Yaw Model

Figure 3.5: Nonlinear Vehicle Roll Model

First the effects of subtracting different pulse forms, frequencies and amplitudes from thesteering wheel input are given, followed by the effects of adding these pulses to the steeringwheel input.

11

3.2 Vehicle dynamics with pulse subtraction

The primary goal of the PASC is to lower the rollover coefficient, but the vehicle trajectoryintended by the driver can not be changed too much by the anti-rollover system if an obstacleon the road needs to be avoided. Therefore, the results of the simulations made for a steeringwheel input with a subtracted pulse are compared to an un-pulsed input, since this gives thedrivers intended uncontrolled vehicle trajectory. The steering wheel angle used for the J-turnmaneuver for the un-pulsed input and for the input with a subtracted pulse can be foundin Figure 3.6. As can be seen, the maximum angle of the steering wheel input is chosen tobe 320 degrees. Taking the steer ratio of 23.5, the total wheel angle becomes 13.6 degrees.The maneuver is performed at a relatively low vehicle velocity of 40 km/h. One might beexpecting that the maneuver is simulated at a lower maximum input and a higher velocity,but it is found that the rollover coefficient for higher velocities shows too much oscillationand the effect of pulse subtraction is therefore less easy to investigate. The used input hasshown to work well for the investigation.

0 2 4 6 8 100

50

100

150

200

250

300

350

time [s]

stee

ring

whee

l ang

le

s [de

g]

Steering wheel input vs time

pulsed inputunpulsed input

Pulse amplitude

Figure 3.6: The pulsed and un-pulsed steering wheel input s for the J-turn maneuver

First the results of subtracting a symmetric pulse will be given in section 3.2.1, followedby subtracting a non-symmetric pulse in section 3.2.2. In section 3.2.3 the rollover coefficientof both pulses will be compared to a steering wheel input with a subtracted constant value.

3.2.1 Symmetric pulse input

The effects of different frequencies using a symmetric pulse is investigated for two pulseamplitudes. In [8] the ratio between the maximum steering angle of the front wheels of theuncontrolled vehicle and the pulse amplitude for the J-turn maneuver is around 5:2. For easeof comparison this ratio is used for the first simulation, resulting in a pulse amplitude of 120degrees. For the second simulation the amplitude is decreased to an arbitrary 80 degrees.

The resulting vehicle trajectory for both amplitudes at different frequencies is shown infigures 3.7 (a) and (b), respectively. The un-pulsed vehicle trajectory is also shown in bothfigures. The following conclusions can be drawn from these figures:

Increasing the frequency from 1 to 4 Hz results in a larger path deviation with respectto the un-pulsed input, independent of the pulse amplitude.

12

A high frequency of 8 Hz results in a smaller path deviation compared to the 4 Hzfrequency.

A higher pulse amplitude results in a larger path deviation with respect to the un-pulsedinput for all frequencies.

The difference in the vehicle trajectory between the frequencies depends on the pulseamplitude.

40 30 20 10 0 10 20 30 4080

70

60

50

40

30

20

10

0Vehicle trajectory

x [m]

y [m

]

unpulsedsym. pulse 1 Hzsym. pulse 2 Hzsym. pulse 4 Hzsym. pulse 8 Hz

(a) Pulse amplitude 120 degrees

40 30 20 10 0 10 20 30 4080

70

60

50

40

30

20

10

0Vehicle trajectory

x [m]

y [m

]

unpulsedsym. pulse 1 Hzsym. pulse 2 Hzsym. pulse 4 Hzsym. pulse 8 Hz

(b) Pulse amplitude 80 degrees

Figure 3.7: Vehicle trajectory for a pulse with an amplitude of 120 and 80 degrees for different frequencies

It is clear that the vehicle trajectory depends on the frequency, but most of all on the pulseamplitude. This is due to the fact that the pulse amplitude is the biggest factor determin-ing the overall average input, since the pulse amplitude determines the amount of degreessubtracted from the steering wheel input.

Note that the vehicle trajectory for pulses with an amplitude of 120 degrees and with afrequency of 1 and 2 Hz are almost the same, but this is not the case for the pulse with the loweramplitude. Hence, it might be concluded that high amplitude pulses with a small frequencyhave almost the same effect on the vehicle trajectory. More investigation is necessary toconfirm this conclusion.

The rollover coefficient for both simulations are shown in figures 3.8 (a) and (b), respec-tively. The following conclusions can be drawn from these figures if one looks at the resultsduring the time the pulse is being subtracted:

Pulses with a frequency of 1 and 2 Hz result in a higher rollover coefficient comparedto the un-pulsed input, independent of the size of the amplitude.

Pulses with a frequency of 4 and 8 Hz result in a lower rollover coefficient compared tothe un-pulsed input and the lowest rollover coefficient is found for the highest frequency,independent of the size of the amplitude.

A higher amplitude results in a lower rollover coefficient for pulses with a frequency of4 and 8 Hz.

13

The difference in the rollover coefficient between the frequencies depends on the pulseamplitude.

0 2 4 6 8 10

0

0.1

0.2

0.3

0.4

0.5

Rollover coefficient vs time

Time [s]

Rol

love

r coe

fficie

nt

unpulsedsym. pulse 1 Hzsym. pulse 2 Hzsym. pulse 4 Hzsym. pulse 8 Hz

(a) Pulse amplitude 120 degrees

0 2 4 6 8 10

0

0.1

0.2

0.3

0.4

0.5

Rollover coefficient vs time

Time [s]R

ollo

ver c

oeffi

cient

unpulsedsym. pulse 1 Hzsym. pulse 2 Hzsym. pulse 4 Hzsym. pulse 8 Hz

(b) Pulse amplitude 80 degrees

Figure 3.8: Rollover coefficients for a pulse with an amplitude of 120 and 80 degrees for different frequencies

A possible explanation of the fact that low frequencies result in a high rollover coefficientis that these frequencies are close to the eigenfrequency of the suspended vehicle body. Thefact that a higher pulse amplitude results in a lower rollover coefficient is due to the factthat for high amplitudes more degrees are subtracted from the steering wheel input. Thisresults in a lower overall average steering wheel input and therefore in a less extreme J-turnmaneuver. This causes a lower rollover coefficient.

Based on all previous results, the following conclusions can be drawn:

The PASC system seems to have good potential to decrease the rollover coefficient,which is in line with the findings of the study done by Kuo.

The rollover coefficient is only decreased for pulses with a specific high frequency.

A pulse with a frequency of 8 hz results in a lower path deviation and in a lower rollovercoefficient compared to a pulse with a frequency of 4 Hz.

The general effects of different frequencies on the vehicle trajectory and rollover coeffi-cient do not depend on the amplitude of the pulse.

Note that the maximum rollover coefficient does not reach the critical value of 1. This isdue to the input used for the simulations and due to the fact that the demo vehicle used inthe software program has a low center of gravity, which makes rollover more difficult. It isalso important to note that the rollover coefficient is only decreased for frequencies above the4 Hz during the time the pulse is being subtracted. To make sure that the rollover coefficientstays beneath a certain threshold, the pulse needs to be applied during a longer period.The exact begin and and time of the pulse subtraction needs to be controlled by a controlsystem. One simulation is performed to study the effect of a longer pulse subtraction time.More information about the performed simulation and the simulation result can be foundin Appendix A. From this result it can be concluded that an increased pulse subtraction

14

time results in a lower rollover coefficient compared to the un-pulsed input during the totalmaneuver time. The differences with the simulation with the shorter pulse-subtraction-timeare very small during this interval. Hence, it appears that the conclusions drawn based onthe previous results do not depend on the time the pulse is subtracted.

3.2.2 Non-symmetric pulse input

Studying the effect of the non-symmetric pulse as described in section 3.1 is done by firstresearching the effect of different frequencies and secondly, by researching the effect of anincreased pulse peak-time value a for a constant frequency. This last research is performedsince it is expected that the peak-time value also has a big effect on the vehicle trajectoryand rollover.

The amplitude for all simulations is chosen to be 120 degrees (also used for the symmetricpulse), since at this amplitude the rollover coefficient is being reduced the most. It can beexpected that the conclusions drawn from these simulation also hold for smaller or biggeramplitudes.

Frequency modulation

To investigate the effect of different frequencies, the peak-time value a (as function of thevibration time T) must be kept constant. For this investigation the value is chosen to be 14 ofthe vibration time. The begin and end points of each pulse need to be the same and almostequal to 0. Therefore, the values b and c as used in (3.4) and (3.5) need to be determined bythe following equation:

f1,i(t = 0) = f2,i(t = T ) = constant 0 for i = 1, 2, 4, 8 Hz (3.8)

Note that this equation only holds for one pulse starting at time t = 0. For a pulse with afrequency of 1 Hz, the value b is chosen to be 0.005. Using this combination of values for aand b the constant value in equation 2.5 is found (3.7267e6) and the value c can now alsobe determined using the same equation. Using (3.4), (3.5) and (3.8), the values b and c asfunction of the frequency can be calculated by:

b =a2

12.5=

1

200f2(3.9)

c =(T a)2

a2b =

9

200f2(3.10)

Using these equations a combination of values a, b and c is found for each frequency (seeAppendix B). These combinations are used for the simulations.

The resulting vehicle trajectories at each frequency are shown in Figure 3.9 (a). Theconclusions drawn from this figure are the same as for the symmetric pulse (see section 3.2.1).The average vehicle trajectories of all the frequencies for the symmetric pulse and for thenon-symmetric pulse are shown in Figure 3.9 (b). It can be seen that the non-symmetricpulse results in a smaller path deviation. This is as expected, since the area above the pulseis lower for the non-symmetric pulse. This means that less is subtracted from the steeringwheel input, causing a higher average steering input and a lower path deviation.

15

40 30 20 10 0 10 20 30 4080

70

60

50

40

30

20

10

0Vehicle trajectory

x [m]

y [m

]

unpulsednonsym. pulse 1 Hznonsym. pulse 2 Hznonsym. pulse 4 Hznonsym. pulse 8 Hz

(a) Vehicle trajectory for a non-symmetric pulsewith different frequencies

40 30 20 10 0 10 20 30 4080

70

60

50

40

30

20

10

0Vehicle trajectory

x [m]

y [m

]

unpulsedaverage symmetric pulseaverage nonsymmetric pulse

(b) Average vehicle trajectory for the symmetricand non-symmetric pulse

Figure 3.9: Vehicle trajectories

The rollover coefficient at each frequency can be found in Figure 3.10 (a). Figure 3.10 (b)shows a zoomed area for clarity. As can be seen, the coefficient is increased for all frequencies.So the non-symmetric pulse results in a lower path deviation with respect to the symmetricpulse, but this specific non-symmetric pulse seems to be unable to decrease the rollover withrespect to the uncontrolled input.

1 2 3 4 5 6

0

0.1

0.2

0.3

0.4

0.5

Rollover coefficient versus time

Time [s]

Rol

love

r coe

fficie

nt

unpulsednonsym. pulse 1 Hznonsym. pulse 2 Hznonsym. pulse 4 Hznonsym. pulse 8 Hz

(a) Unzoomed

2 2.5 3 3.5 4 4.5 50.35

0.4

0.45

0.5

0.55Rollover coefficient versus time

Time [s]

Rol

love

r coe

fficie

nt

unpulsednonsym. pulse 1 Hznonsym. pulse 2 Hznonsym. pulse 4 Hznonsym. pulse 8 Hz

(b) Zoomed

Figure 3.10: Rollover coefficient for the non-symmetric pulse with an amplitude of 120 degrees

Pulse peak-time modulation

To determine the effect of the peak-time value a, one simulation is made using a pulse witha value of a = 34T . This means that the pulse now consists of a smooth, gradually decreasingslope combined with a sharp, gradually increasing slope. The simulation is performed for apulse with an amplitude of 120 degrees and a frequency of 8 Hz, since this high frequencyresults in the lowest rollover coefficient. Note that changing the value a also implies a changein the values of b and c.

16

The vehicle trajectory and the rollover coefficient for this simulation is shown in figures3.11 (a) and (b), respectively. The vehicle trajectory and the rollover coefficient for thesymmetric pulse and for the non-symmetric pulse with the old value of a = 14T are addedfor comparison. As can be seen, a pulse with a high peak-time value results in a slightlylarger path deviation with respect to a pulse with a low peak-time value. However, therollover coefficient is significantly lower for the pulse with a high peak-time value. Hence, itcan be concluded that a pulse with a high a-value seems to have good potential to decreasethe rollover coefficient combined with a small path deviation. However, the symmetric pulseshows the lowest rollover coefficient, although it also shows a larger path deviation.

40 30 20 10 0 10 20 30 4080

70

60

50

40

30

20

10

0Vehicle trajectory

x [m]

y [m

]

(a) Vehicle trajectory

0 2 4 6 8 10

0

0.1

0.2

0.3

0.4

0.5

Rollover coefficient versus time

Time [s]

Rol

love

r coe

fficie

nt

unpulsednonsym. pulse, a = 1/4 Tnonsym. pulse, a = 3/4 Tsym. pulse

(b) Rollover coefficient

Figure 3.11: Vehicle trajectory and rollover coefficients for different pulse forms

So far the effects of different pulse forms on the vehicle trajectory and rollover are inves-tigated separately. The next step is to combine the two and to investigate which pulse formcan best be subtracted from the steering wheel input to decrease the rollover coefficient themost. This investigation is performed next.

3.2.3 Optimal subtraction method

For a good comparison between the rollover coefficient of each pulse form the vehicle trajectoryneeds to be the same for all. The same vehicle trajectory can be obtained by modifying theamplitude of each pulse. The investigation is only performed for pulses with a frequency of8 Hz, since this high frequency decreases the rollover coefficient the most. It is found thatdecreasing the amplitude of the symmetric pulse from 120 to 76 degrees gives the same vehicletrajectory as the non-symmetric pulse with an amplitude of 120 degrees and a peak-time valueof 34T (see section 3.2.2). The simulation results are also compared to the rollover coefficientobtained from a steering wheel input with a constant (un-pulsed) subtracted value. Thismakes it possible to investigate if the PASC system is better able to decrease the vehiclerollover than the ASC system.

Before a proper comparison can be ensured it needs be noted that one of the reasonsthe non-symmetric pulse used in section 3.2.2 results in a relatively high rollover coefficientcan be that the falling and rising slope of the pulse are too sharp. A second reason can bethat the non-symmetric pulse used is C0 continuous. ADAMS might not be able to workwell with a C0 continuous pulse, probably due to interpolation problems. A C0 continuous

17

pulse can also be difficult to produce by the pulse actuation design as described in Chapter 5.Therefore, a simulation is made for a steering wheel input with a subtracted C1 continuousnon-symmetric pulse with a slightly less sharp falling and rising slope. This new pulse is givenby the following equation:

f(t, T ) =A

2(1 cos(eq(Tmod(t,T )

n) 1)) (3.11)

with A the amplitude of the pulse, T the vibration time of the pulse, t the time and

q =ln(2pi + 1)

T n(3.12)

n = 0.335 (b

a+ 0.46) (3.13)

The peak-time value is given by the factor ba. A representation of this pulse with a peak-

time value of 34T can be found in Figure 3.12. The C0 continuous non-symmetric pulse with

the same peak-time value and the symmetric pulse are also shown for comparison.

0 0.2 0.4 0.6 0.8 11

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

time

y

symmetric pulseC0 continous nonsym. pulseC1 continous nonsym. pulse

Figure 3.12: Representation of the different pulse forms

For the simulation the peak-time value of this new pulse form is chosen at 34T , since thispeak-time value results in the lowest rollover coefficient for the C0 continuous non-symmetricpulse (see section 3.2.2.). The amplitude of the pulse is chosen such that the vehicle trajectoryis the same as the inputs with the two other subtracted pulse forms and the same as the inputwith the constant subtracted value.

The resulting rollover coefficients for the different inputs are given in Figure 3.13. Ascan be seen in this figure, the new non-symmetric pulse results in a significant decrease inthe rollover coefficient with respect to the other non-symmetric pulse and even shows a lowercoefficient than the symmetric pulse. So the new non-symmetric pulse studied so far has thehighest potential to decrease the rollover coefficient compared to other pulses. However, thelowest rollover coefficient is found for the input with the constant subtraction.

It can be seen that all pulse forms oscillate around the input with a constant subtractedvalue. Increasing the peak-time value of the non-symmetric pulses or increasing the pulse

18

0 2 4 6 8 10

0

0.1

0.2

0.3

0.4

0.5

Rollover coefficient versus time

Time [s]

Rol

love

r coe

fficie

nt

unpulsednonsym. pulse, a = 3/4 Tsym. pulse, amp 76new nsp, amp 72constant subtraction

Figure 3.13: Rollover coefficient for inputs with different subtracted pulses and for an input with a constantsubtracted value

frequency even more than 8 Hz can possibly result in a decreased rollover coefficient, but itwill always be higher than the input with the constant subtracted value. This means that it isbest to subtract a constant value instead of a pulse from the steering wheel input to decreasethe chance of rollover of the vehicle. Hence, the ASC system works better than the PASC.

The effect of adding the different pulse forms to the steering wheel angle is investigatednext.

3.3 Vehicle dynamics with pulse addition

Due to understeer or wind disturbance for example, the vehicle can deviate from a desiredtrajectory. In section 3.2 it is found that modifying the amplitude of each different subtractedpulse results in a specific vehicle trajectory. This means that for a steering wheel input withan added pulse a specific vehicle trajectory can also be obtained by modifying the amplitude.Hence, adding a pulse with a specific amplitude can decrease or even delete a path deviation.Therefore, the PASC system can be used for track following. Section 3.2 shows furthermorethat subtracting different pulse forms with a certain frequency results in a lower rollovercoefficient compared to the uncontrolled input. So, adding the pulse will consequently resultin a higher rollover coefficient.

Two questions arise from above observations: First, which pulse form can best be used fortrack following without increasing the rollover coefficient too much and second, if adding apulse to the steering wheel is better than adding a constant (un-pulsed) value to the steeringwheel input.

To investigate these questions the effect of adding different pulse forms on the rollovercoefficient is analyzed and compared to the steering wheel input with a constant added value.It is not expected that the results change significantly with respect to subtracting the pulseand therefore only the different pulses at 8 Hz are being compared. The simulated drivingmaneuver is again the J-turn maneuver. A representation of the vehicle steering wheel inputused for this maneuver can be found in Figure 3.14 (a). The vehicle speed is chosen to be 40km/h, as is used for all earlier performed simulations. As already noted, the vehicle modelused in ADAMS has understeer and therefore the un-pulsed trajectory is not the desired

19

trajectory. The desired vehicle trajectory is determined at a speed of 3.6 km/h, since atlow vehicle speeds the influence of the understeer coefficient on the vehicle trajectory can beneglected. The maximum uncontrolled steering wheel input is chosen to be 120 degrees. Thedesired vehicle trajectory and the un-pulsed vehicle trajectory are shown in Figure 3.14 (b).

0 2 4 6 8 100

20

40

60

80

100

120

140

160

180

time [s]

stee

ring

whee

l ang

le

s [de

g]

Steering wheel input vs time

pulsed inputunpulsed input

Pulse amplitude

(a) Steering wheel angle input

10 0 10 20 30 40 50 6080

70

60

50

40

30

20

10

0Vehicle trajectory

x [m]y

[m]

desired trajectoryunpulsed trajectory

(b) Vehicle trajectory

Figure 3.14: The un-pulsed and pulsed steering wheel angle input and vehicle trajectory

The amplitude of each different pulse form is now determined such that the steering wheelinput with the added pulse gives the desired trajectory. It is found that the amplitude of thesymmetric pulse needs to be 25 degrees, the amplitude of the C0 continuous non-symmetricpulse 51 degrees and the amplitude of the C1 continuous non-symmetric pulse needs to be23 degrees to get the desired trajectory. So using these pulses with these amplitudes resultsin the black line visible in figure 3.14 (b). Note that the peak-time value is 34T for bothnon-symmetric pulses. The rollover coefficient for the different pulse forms and for the inputwith a constant added value are shown in Figure 3.15.

0 2 4 6 8 100.05

0

0.05

0.1

0.15

0.2

0.25

0.3Rollover coefficient versus time

Time [s]

Rol

love

r coe

fficie

nt

unpulsedC0 cont. nonsym. pulseC1 cont. nonsym. pulsesymmetric pulseaveraged unpulsed

Figure 3.15: Rollover coefficient for different pulse forms and for the constant added value

Comparing the different pulse forms shows that the rollover coefficient is the lowest for thesymmetric pulse, closely followed by the C1 continuous non-symmetric pulse. The steeringwheel input with a constant added value results in the lowest rollover coefficient. The rollover

20

coefficient obtained from the steering wheel input with the different added pulses oscillatesagain around the steering wheel input with the constant added value. So changing the peak-time of the C1 continuous non-symmetric pulse or increasing the frequency will result in adifferent rollover coefficient (as can also be seen in section 3.2.2), but the rollover coefficientwill not be lower than the steering wheel input with a constant added value.

Note that, although the rollover coefficient in this case does not reach the value of 1, thepulse can not always be added. If, due to a certain input, the rollover coefficient reaches avalue close to 1 and a pulse with a certain high amplitude is added, the vehicle will roll over.The path deviation can possibly still be decreased slightly, but can not be deleted. The inputwith a constant added value will be able to decrease the path deviation the most, since itincreases the rollover coefficient the least.

3.4 Discussion

All the simulation results lead to the conclusion that the PASC system is able to decrease therollover coefficient by subtracting a pulse and is able to decrease or even delete a path deviationby adding the pulse to the steering wheel input. A C1 continuous non-symmetric pulse with apeak-time value of (3)(4)T has the best potential to decrease the rollover coefficient comparedto other pulses. The exact vehicle trajectory and rollover coefficient depend on the form,frequency and amplitude of the pulse. Subtracting a pulse with a high frequency seems toresult in the best combination of rollover coefficient and vehicle trajectory. However, the bestinput is found to be the one with a subtracted or added constant value, since this results inthe lowest rollover coefficient. This means that the ASC system works better than the PASCsystem.

For the purpose of the study described in this report a J-turn maneuver is performed,whereas different driving maneuvers might have different reactions. Therefore more researchcan be conducted to shed light on the effects of different driving maneuvers.

The vehicle model used for the simulations does not resemble a SUV. Since the major goalof the PASC system is to decrease the vehicle rollover of especially SUV, further research canbe performed using a vehicle model which resembles a SUV more.

C.C. Kuo [8] has shown that a steering wheel input with a subtracted symmetric pulseresults in a rollover coefficient with some bumps higher than the uncontrolled vehicle rollovercoefficient. The bumps found can be due to the fact that the symmetric pulse used has aC0 continuity at the beginning and end of each pulse. He has shown that the C0 continuousnon-symmetric pulse is able to eliminate these bumps and to decrease the vehicle rollover,but the results given in this report show that the non-symmetric pulse is not able to decreasethe rollover coefficient below the uncontrolled vehicle rollover coefficient. This can be duetoo a too sharp falling and rising slope of the pulse or due too interpolation problems inthe ADAMS software. More research can be performed upon this subject. Kuo also statesthat the overal reduction in its total amount is smaller for the C0 continuous non-symmetricpulse with respect to the symmetric pulse, which matches the findings of this report more.From the results given in this chapter it is clear that the pulse presented in [8] is not able toreduce the rollover coefficient as much as the new non-symmetric pulse proposed or even thesymmetric pulse.

Since different pulse forms give different results, it might be possible that there is a certainform which is able to decrease the rollover coefficient even more than the different forms used

21

in this report. It might also be that there is a certain pulse form which is able to decreasethe rollover coefficient even more than subtracting a constant value. However, this is notexpected since the rollover coefficient of all the pulses oscillate around the rollover coefficientgiven by the input with a constant added or subtracted value.

Since the tyres are moving sideways over the ground, it is expected that the PASC systemwill result in excessive wear of the tyres. It is also expected that the system has a negativeeffect on the ride comfort, since rolling of the vehicle with a certain frequency can be annoyingfor the passengers.

To make sure the found results are acceptable, the simulation results need to be validated.This validation is described in the next chapter.

22

Chapter 4

Results validation

To check whether the results obtained with the software program ADAMS as shown in Chapter3 are acceptable, the simulation results are validated by comparing simulation results fromADAMS with simulation results obtained using the software program Maple in combinationwith DynaFlexPro (DFP).

First information about the new software programs and about the performed simulationsis given, followed by the simulation results. At the end a conclusion based upon these resultsis given.

4.1 DFP and Maple simulations

The sofware program Maple is able to compute and manipulate symbolic expressions. Thesymbolic expressions, the kinematics and dynamic equations of a system, can be automaticallygenerated by the program DFP wherein the studied system model is built. The model hasfourteen degrees of freedom: six for the chassis (three translational and three rotational), onefor the spin of each tire and one for the vertical prismatic joint of each suspension link. Theinput used in this program is given by the steer angle of the front wheels.

One of the goals of the PASC is to reduce the vehicle rollover of (especially) SUVs.Therefore, the pulse actuation system, as described in Chapter 4, needs to be designed fora SUV. For the design it needs to be known what the maximum amount of wheel angle ofthe front wheels is before a SUV starts to rollover. As already noted, for the simulationsperformed in Chapter 3 a vehicle model is used which does not resemble a SUV. However,the parameters of the vehicle model already implemented in DFP are parameters from theChevrolet Equinox, which is a SUV. Therefore, DFP in combination with Maple is first usedto determine this maximum amount of wheel angle input. This information will also be usedfor the validation of the ADAMS vehicle model.

For this first investigation the J-turn maneuver as described in section 2.1 is simulated ata vehicle velocity of 80 km/h. This velocity is chosen since it is more likely that the vehiclerolls over at this high velocity compared to the relatively low velocity of 40 km/h used forthe simulations in Chapter 3. It is found that an input on the front wheels above 3 degreesresults in rollover of the Equinox model. This maximum steer angle of the front wheels ischosen as input for the simulations in Maple. Using the steer ratio of 23.5 found in section2.1, the maximum steering wheel input for the ADAMS simulations becomes 70.5 degrees.

Simulations are performed using a steering wheel input with a subtracted symmetric pulse

23

and a subtracted C0 continuous non-symmetric pulse with a frequency of 1 and 4 Hz. Onemight expect a frequency of 8 Hz, because this results in the lowest rollover coefficient, but theinput for this simulation in Maple would be too extensive. The used frequencies are expectedto work well for the validations and it is expected that the results found also hold for otherfrequencies. The pulse amplitude on the front wheels is in this case chosen to be 1.2 degrees.This gives the same ratio (5:2) between the maximum wheel angle input and the amplitudeof the pulse as also used for the simulations in Chapter 3. Using the steer ratio of 23.5 thisresults in a pulse amplitude on the steering wheel of 28.2 degrees.

The results validation consist of comparing the self-aligning moment of the front wheels(Mz). The self-aligning moment is caused by the lateral force of the tire produced by the slipangle. The force acts through a point behind the center of the wheel, the pneumatic trail, ina direction such that it attempts to re-align the tire. The total self-aligning moment of thefront wheels can be calculated by adding the self-aligning moment of the left wheel with theself-aligning moment of the right wheel.

For a good comparison the vehicle parameters used in the program ADAMS (as given inTable 2.1) are implemented in the vehicle model built in DFP. So the model used in DFPresembles the vehicle model used in ADAMS as good as possible.

4.2 Simulation results

The self-aligning moment for the input with a subtracted symmetric pulse with a frequencyof 1 Hz and 4 Hz given by both programs are presented in figures 4.1 (a) and (b), respec-tively. The self-aligning moment for the input with a subtracted non-symmetric pulse with afrequency of 1 Hz and 4 Hz are presented in figures 4.2 (a) and (b), respectively. As can beseen in these figures, the self-aligning moment given by both software programs is negative.This is due to the fact that the J-turn is performed to the right: the self-aligning moment actscounterclockwise and since a moment clockwise is taken as positive, the resulting self-aligningmoment is negative. Furthermore it can be seen that the minimum of the self-aligning mo-ment is lower for the DFP model than for the ADAMS model. This is due to the differencein the models. The most important difference between the models is the difference in sus-pension: in DFP the model used has a (simplified) vertical suspension, while ADAMS uses aMcPherson suspension. This can also explain the fact that the self-aligning moment in theADAMS results go further back to zero during the pulse.

4.3 Discussion

The self-aligning moment is compared, because at first it was expected that this self-aligningmoment can be directly related to the applied torque on the steering wheel. The appliedtorque on the steering wheel to turn the front wheels is information needed to design thepulse actuation system proposed in Chapter 4. However, it is found that the torque on thesteering wheel depends on geometric parameters of the steering system [10]. These geometricvariables are unknown and therefore it is not possible to relate the self-aligning momentdirectly to the torque on the steering wheel.

The results obtained by both software programs show a distinctive difference, but thesedifferences can be explained by the difference in the models used. The differences seem to be

24

0 1 2 3 4 5 6 7 8100

80

60

40

20

0

20Selfaligning moment vs time

Time [s]

Mz

[Nm]

ADAMSMaple

(a) 1 Hz

0 1 2 3 4 5 6 7 8100

80

60

40

20

0

20Selfaligning moment vs time

Time [s]M

z [N

m]

ADAMSMaple

(b) 4 Hz

Figure 4.1: Self-aligning moment in ADAMS and Maple for a symmetric pulse with a frequency of 1 Hz and4 Hz

0 1 2 3 4 5 6 7 8100

80

60

40

20

0

20Selfaligning moment vs time

Time [s]

Mz

[Nm]

ADAMSMaple

(a) 1 Hz

0 1 2 3 4 5 6 7 8100

80

60

40

20

0

20Selfaligning moment vs time

Time [s]

Mz

[Nm]

ADAMSMaple

(b) 4 Hz

Figure 4.2: Self-aligning moment in ADAMS and Maple for a non-symmetric pulse with a frequency of 1 Hzand 4 Hz

25

consistent for different pulse forms and for different frequencies. From this it can be concludedthat the results found in Chapter 3 can be accepted.

The information obtained in this chapter and in Chapter 3 can now be used to design thepulse actuation system for the setup.

26

Chapter 5

Pulse actuation system

The results of the performed simulations given in chapter 2 show that the PASC system hasgood potential to lower the vehicle rollover. To study the mechanical effect of the PASCsystem on the total mechanical steering system and to perform experiments for the validationof the results found in Chapter 3 and 4 a test setup needs to be built. This setup will consistsof a steering column, steering rack, a set of wheels and a pulse actuation system. This pulseactuation system adds or subtracts the pulse to the drivers steering wheel input and will beplaced between the steering wheel column and the steering pinion/rack. The design of thepulse actuation system is described in this chapter.

The pulse actuation system consists of a gear-train assembly1 and a pulse actuator. Thedesign of the gear-train assembly and the choice of gear teeth will be described first. Second,different pulse actuators will be discussed and based upon the advantages and disadvantagesa pulse actuator will be chosen. Since a high frequency results in a lower rollover coefficient,the chosen actuator needs to be further designed such that an optimized maximum pulsefrequency can be added or subtracted to the drivers steering wheel input. For the design itis necessary to obtain the maximum torque and power needed to generate the pulse motionsof the front wheels as described in Chapter 3 and 4, so this is studied beforehand.

5.1 Gear-train assembly

The gear train assembly is designed taking into account the following constraints:

The driver does not feel the pulse on the steering wheel.

The ratio between the steering wheel input and output of the pulse actuation system is1:1 if no pulse is applied.

The steering wheel input and output are co-linear, which means that the input andoutput are aligned.

The rotational directions of the input and output are the same.

The added or subtracted pulse frequency (with a specific amplitude) needs to be as highas possible.

1A first setup of this assembly has been designed by Alexander Berlin, a student at the University ofWaterloo

27

Figure 5.1 (a) shows a 3-dimensional drawing and Figure 5.1 (b) shows the working scheme ofthe assembly. As can be seen, the assembly consists of 4 spur gears and a planetary gear-set.Gear 1 is connected to the steering wheel and is the input of the assembly. The gears 2 to 4are necessary to make the steering wheel input versus the output of the total system 1:1, ifno pulse is applied. The planetary gear-set consists of the sun (gear 8), the ring (gear 7) andthree planets (gears 6) connected to the carrier (gear 5). The carrier is directly connected togear 4. The sun-gear is connected to the steer-rack and is the output of the assembly. Thepulse will be applied on the ring gear by the pulse actuator. Details about the pulse actuatorcan be found in section 4.4.

(a) 3-Dimensional drawing designed by A. Berlin (b) Working scheme

Figure 5.1: Gear-train assembly design

Equations belonging to the system indicated in Figure 5.1 are:

R11 = R22 (5.1)

2 = 3 (5.2)

R33 = R44 (5.3)

4 = 5 (5.4)

8 = (z + 1)5 z7 (5.5)

R7 = R8 + 2R6 (5.6)

T7 = zT8 (5.7)

T5 = (z + 1)T8 (5.8)

With Ri the radius of gear i, i the rotational speed of gear i, z the ratio between the ring-gear and sun-gear (z = R7

R8) and Ti the torque on gear i.

When the ring gear is stationary (7 = 0), the input versus output (1 : 8) has to be 1:1.Taking this into account the spur gears need to be chosen such that the following equation,found by using equations (5.1) to (5.5), holds:

z + 1 =R2R4

R1R3(5.9)

To make the steering wheel input and planetary gear-set input co-linear, the following equationhas to hold as well:

R1 +R2 = R3 +R4 (5.10)

28

The number of teeth of each gear can now be chosen such that all the above equationshold, but it needs to be taken into account that the gears of the gear train assembly needto be provided by the company Boston Gears [11]. The system also has to be as cheap aspossible and as compact as possible and the gears need to be able to withstand the maximumapplied torque and power. These last two constraints depend on the pressure angle, numberof teeth and diametral pitch of the gears. These three are described below.

Pressure angle

The pressure angle is the angle at a pitch point between the line of pressure which is normalto the tooth surface and the plane tangent to the pitch surface. The company supplies gearswith pressure angles of 14.5 and 20. Gears with a higher pressure angle have a higher loadcarrying capacity, but gears with a lower pressure angle are better for extensive use, haveless change in backlash and have a higher contact ratio and therefore a smoother and quieteroperation [11]. Because of this a pressure angle of the gears of 14.5 is chosen.

Number of teeth

The ratio between the ring-gear and the sun-gear (z) is chosen to be 1.5. A lower ratio willresult in a bigger total gear diameter and therefore violates the compact constraint. Takinga ratio of 1.5 and using the gears provided by the company, the smallest diameter of gears isfound taking 48 teeth for the sun-gear, 12 teeth for the planets and 72 teeth for the ring-gear.The number of teeth of gears 1 to 4 can be chosen such that equations 5.9 and 5.10 hold.This results in 16 teeth for gear 1, 20 teeth for gear 2, 12 teeth for gear 3 and 24 teeth forgear 4.

Diametral pitch

The gear supplier provides gears not only with different numbers of teeth (N), but also witha a different diametral pitch (P). The diametral pitch is the number of teeth in the gear foreach inch of pitch diameter. Both variables determine the pitch diameter (D) of a gear bythe following equation:

D =N

P(5.11)

The diametral pitch is an important factor determining the maximum torque and power thatthe gear can handle: the higher the diametral pitch, the lower the maximum torque and powerthat the gear can withstand.

The maximum torque and power supplied to the gears depend on the motor driving thepulse actuator. The pulse actuator will be driven by a motor available at the University ofWaterloo. This available motor is the Kollmorgen Seidel 6SM47L-3000. The rated speed ofthis motor is 3000 rpm, the rated torque at this rated speed is 2.2 Nm and the rated power is690 W. More motor characteristics can be found in Appendix C. Using the rated power andthe approximated horsepower and torque ratings provided by the gear supplier, it is foundthat the diametral pitch of the gears has to be 12 or less, otherwise the smallest gears (theplanets on the planetary gear-set) will not be able to withstand the supplied power. Since asmaller diametral pitch results in an undesired bigger diameter of the gears a diametral pitchof 12 is chosen for now. The maximum torque and power that a gear can handle does not

29

only depend on the diametral pitch, but also on the rotational speed of the gear. At a lowerspeed the gear can handle a lower power, but a higher torque. The rotational speed and themaximum torque supplied to each gear depends on the pulse actuator described in the nextsection. At the end of this chapter it will be proven that a diametral pitch of 12 for each ofthe gears is big enough to withstand the torque and power supplied on each gear separately.

5.2 Pulse actuator

The results given in Chapter 3 show that the rollover coefficient and vehicle trajectory dependon the frequency and the amplitude of the pulse. They also show that the rollover coefficientcan be decreased by subtracting the pulse and that a desired trajectory can be obtained byadding the pulse to the steering wheel input. Taking this into account the pulse actuator mustbe able to modulate both the frequency and the amplitude of the pulse and it must be ableto switch between adding and subtracting of the pulse from the steering wheel input. Themechanism must be able to satisfy these constraints with as little motor control as possible.

First a study is performed using the books written by I. I. Artobolevsky [12] to see if thereis an existing mechanism able to satisfy the above constrains. Some of these mechanisms aredescribed below. Based on the found information a mechanism is chosen. This mechanismis further designed (see section 4.4) to be able to apply a pulse with a frequency as high aspossible.

Mechanisms

A mechanism able to modulate the frequency relatively easily by changing the rotational speedof the motor is the three-bar mechanism (see Figure 5.2 (a)) and the four-bar mechanism (seeFigure 5.2 (b)). In the three-bar mechanism link 1 rotates around fixed axis A, causing link2 to oscillate around fixed axis B. In the four-bar mechanism link 1 rotates around fixed axisA, causing link 3 to oscillate around fixed axis D.

(a) three-bar rotating-slotted-link mechanism[12]

(b) four-bar crank and rocker-arm mechanism[12]

Figure 5.2: multiple-bar mechanisms

One of the disadvantage of these multiple-bar mechanisms is that the angle of oscillationcan not be changed and therefore, the amplitude of the movement can not be adjusted without

30

motor control. There are some mechanisms that are able to change the angle of oscillationwithout too much motor control. These are shown in Figure 5.3.

In the mechanism shown in Figure 5.3 (a), link 2 has collar b encircling eccentric 1, whichrotates around fixed axis A. The stroke of link 3 or the oscillation of link 2 can be changedwith screw 4 by adjusting the distances between axis A and the center of roller B.

In the mechanism shown in Figure 5.3 (b) the input is given by the disk rotating aroundfixed axis A, causing link 1 to oscillate around fixed axis C. The length C-D of rocker arm 1can be changed by turning screw 2, thereby changing the angle of oscillation of link 1.

In Figure 5.3 (c) the input of the mechanism is given by crank 1 rotating around fixedaxis C, causing link 4 to slide in guide c and causing rocker link 6 to oscillate around axisE. The angle of rotation can be varied by changing the position of point B by screw a. Notethat in this figure a ratchet is drawn, but link 6 can also be connected directly to wheel 7.

The input of the mechanism shown in Figure 5.3 (d) is given by crank link 1, rotatingaround fixed axis B and is connected to slotted link 5 at point C. The rocker link is link 2and oscillates around fixed axis B when link 1 rotates. The angle of oscillation of link 2 canbe changed by changing the position of pin A by screw 3.

(a) Link-gear mechanism with driven link angleof oscillation adjustment [12]

(b) Lever-ratchet mechanism with variableangular velocity of the ratchet wheel [12]

(c) Four-bar mechanism with a rocker arm ofvariable length [12]

(d) Three-bar link-gear mechanism with drivenlink stroke adjustment [12]

Figure 5.3: adjustable-amplitude mechanisms

The frequency of the movements of these mechanisms can also be easily changed bychanging the rotational speed of the motor. The disadvantage of these systems is however

31

that a lot of motor control is required to change from adding to subtracting of the pulse fromthe steering wheel input. Since a lot of motor control is necessary to make sure that all theconstraints are satisfied, a simple gear-mechanism is chosen for applying the pulse on thering-gear of the planetary gear-set.

Worm-gear mechanism

To drive the ring-gear one can take either a spur-gear or a worm-gear. A big disadvantageof driving the ring-gear with a spur-gear is that the ring-gear will not be stationary whenthe driver rotates the steering wheel, because the available motor is not equipped with abrake. This is why it is chosen to drive the ring-gear with a worm-gear. Other advantages ofworm-gears are that they are smooth and quiet, have a high ratio speed reduction and requirelimited space.

It needs to be taken into account that the gears shall not be expected to hold a load whenthe worm-gear is at rest, but theoretically a worm-gear will not back drive if the friction angleis larger than the worm lead angle [11].

The ratio between the worm-gear and the ring-gear determines the maximum pulse fre-quency which can be applied to the steering wheel column. To choose the ratio it needsto be known how much torque and power is necessary to generate a specific pulse motion.Therefore, this is investigated next.

5.3 Power/Torque calculation

The torque and power needed to generate a specific pulse can be given directly by the softwareprogram ADAMS, but most of these results are unreliable, because they do not converge. Sodecreasing the integration tolerance of the software program gives different simulation results.This is possibly due to interpolation problems. Therefore, an analytical model is used todetermine this maximum torque and power. The analytical model is described first, followedby the analytical results.

Analytical model

The analytical model is derived using the simplified steering system as shown in Figure 5.4.As can be seen, the steering system consist of a steering wheel and column, a pinion, a

rack and two tires. Ieq, beq, keq give the equivalent inertia, damping and stiffness of the totalsteering system. The system can be described by the following differential equation:

Ts = Ieq s + beq s + keqs +1

isMz Tps (5.12)

with Ts the torque needed to rotate the steering wheel, s the acceleration of the steeringwheel, s the angular velocity of the steering wheel and s the angle of the steering wheel,r the scale factor to account for torque reduction by the steering gear and Tps the torquedelivered by the power steering system. The influence of the equivalent damping and stiffnessis found to be very small compared to the equivalent inertia and can therefore be neglected.The self-aligning moment generated by the front tires is counteracted by the power steering.The differential equation can now be reduced to the following equation:

Ts = Ieq s (5.13)

32

Figure 5.4: Steering system

The power needed to rotate the steering wheel can be calculated by the equation:

Ps = Tss (5.14)

To calculate the equivalent inertia of the system, some converged results from ADAMSsimulations are used. The simulations consist of the J-turn maneuver at a vehicle speed of80 km/h. The maximum steering wheel angle input is taken to be 120 degrees from which asymmetric pulse with an amplitude of 48 degrees is subtracted. The simulations are performedusing different frequencies. The results are:

1 Hz converges to a maximum torque on the steering wheel of 1.7 Nm

2 Hz converges to a maximum torque on the steering wheel of 6.78 Nm

3 Hz does not really converge, but shows an approximate maximum torque on thesteering wheel of 15 Nm

Using these converged results and (5.13) an equivalent inertia of the total system is found tobe 0.051 kgm2.

To validate (5.13) with the found equivalent inertia, the torque on the steering wheelduring several repeated pulses given by the ADAMS simulations is set against the accelerationof the steering wheel. The results are shown in Figure 5.5. As can be seen in this figure,the simulation results are not exactly the same as the symbolic calculation, although they dooscillate around it. The maximum and minimum torque on the steering wheel lie exactly onthe symbolic calculation line. Hence, (5.13) gives a good approximation of the needed torquefor the simulated pulse at these 3 frequencies and is thereby validated.

Analytical results

The maximum torque and power on the steering wheel needed to generate one pulse arecalculated for all three pulse forms as used in Chapter 3. For both non-symmetric pulses twodifferent peak-time values, 23T and

34T , are used. This is done to see if the peak-time value has

a big effect on the torque and power. First the torque and power are set against the frequency

33

400 300 200 100 0 100 200 300 40025

20

15

10

5

0

5

10

15

20

25

acceleration of steering wheel [rad/s2]

Torq

ue o

n st

eerin

g wh

eel [N

m]

acceleration vs torque

3 Hz2 Hz1 Hzanalytical

Figure 5.5: Torque versus steering wheel acceleration

for each different pulse form with an amplitude of 28.2 degrees (as described in Chapter 3).The results are shown in Figure 5.6. As can be seen, the maximum torque and power neededto generate the non-symmetric pulses rise a lot faster compared to the symmetric pulse. Thehighest needed torque and power are found for the C0 continuous non-symmetric pulse withthe high peak-time value. The lowest needed torque and power are found for the symmetricpulse. This can be explained by the fact that the acceleration of the symmetric pulse is lowerthan for a non-symmetric pulse. Increasing the peak-time value of the non-symmetric pulsesresults in a fair amount of increased torque and power, especially for higher frequencies.

The maximum needed torque and power do not only depend on the frequency but alsoon the amplitude of the pulse. Therefore, the maximum needed torque and power have alsobeen plotted versus the amplitude for each different pulse form. This is done for a frequencyof 1, 2, 4 and 8 Hz as also used for the simulations in Chapter 3. For the non-symmetricpulses this is done again for the two different peak-time values. The torque and power versusthe amplitude for the symmetric pulse, the C1 continuous non-symmetric pulse and the C0

continuous non-symmetric pulse are shown in Figures 5.7, 5.8 and 5.9, respectively. In allthe figures it can be seen that the maximum needed torque and power rise rapidly when theamplitude is increased, especially for high frequencies.

All the above results show that the maximum torque and power needed for the pulsedepend strongly on the used pulse form, frequency and amplitude. In Chapter 3 it hasbecome clear that the rollover coefficient is slightly lower for the C1 continuous non-symmetriccompared with the symmetric pulse at a frequency of 8 Hz, but the needed maximum torqueand power for this non-symmetric pulse is a lot higher. Because of this one might considerusing the symmetric pulse as input on the steering column instead of the C1 continuousnon-symmetric pulse. Therefore, the symmetric pulse is used to further design the pulseactuator.

34

0 1 2 3 4 5 6 7 8 9 100

200

400

600

800

1000Torque

[Nm]

symmetric pulseC1 nonsym. pulse, peak time = 3/4 TC1 nonsym. pulse, peak time = 2/3 TC0 nonsym. pulse, peak time = 3/4 TC0 nonsym. pulse, peak time = 2/3 T

0 1 2 3 4 5 6 7 8 9 100

2000

4000

6000

8000

10000Power