Evaluation of a Pulsed Active Steering Control System

Transcription

1 Evaluation of a Pulsed Active Steering Control System R.Vos DCT 29.1 Traineeship report Coach: Prof. J. McPhee Supervisor: Prof.dr. H. Nijmeijer Technische Universiteit Eindhoven Department Mechanical Engineering Dynamics and Control Group Eindhoven, February, 29

2 Acknowledgements I would like to thank my supervisors, Prof. J. McPhee and Prof. A. Khajepour for the opportunity to do this internship at the University of Waterloo and for their support, guidance and knowledge. I would also like to thank A. Abdel-Rahman for all his help and support throughout the internship 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 encouragement to make this achievement possible. i

3 Abstract In this report the effect of a Pulsed Active Steering Control system (PASC) on a vehicle trajectory and rollover is studied. Former studies have shown that this system is able to prevent rollover better than the Active Steering Control system, the Direct Yaw Moment Control system and the Integrated Control system. However, different pulse forms, frequencies and amplitudes show different effects on the vehicle trajectory and rollover. These effects are investigated in more detail in this report by simulating J-turn maneuvers using a standard vehicle with the software program ADAMS. The vehicle trajectory is directly given by the program, whereas the vehicle rollover is investigated by studying the rollover coefficient. The primary goal of the PASS is to decrease the vehicle rollover and therefore, simulations are performed using a steering wheel input with a subtracted pulse. The secondary goal is to use the 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 have a big effect on the vehicle trajectory and rollover coefficient. A high frequency reduces the rollover 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 reduce the rollover coefficient below a certain threshold. A C 1 continuous non-symmetric pulse is able to reduce the rollover coefficient the most compared to a symmetric pulse and a C 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 and DynaFlexPro. 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 of a gear-train assembly and a pulse actuator. The gear-train assembly comprises 4 spur gears and a planetary gear-set. A worm-gear is taken as pulse actuator. All the gears of the system are chosen such that a pulse with a maximum frequency can be applied to the steering wheel column and such that they can handle the torque and power supplied by an available motor. ii

4 List of Figures 3.1 Vehicle motions defined according to the SAE convention Representation of the J-turn maneuver input Representation of the used symmetric and non-symmetric pulse Nonlinear Vehicle Yaw Model Nonlinear Vehicle Roll Model The pulsed and un-pulsed steering wheel input δ s for the J-turn maneuver Vehicle trajectory for a pulse with an amplitude of 12 and 8 degrees for different frequencies Rollover coefficients for a pulse with an amplitude of 12 and 8 degrees for different frequencies Vehicle trajectories Rollover coefficient for the non-symmetric pulse with an amplitude of 12 degrees Vehicle trajectory and rollover coefficients for different pulse forms Representation of the different pulse forms Rollover coefficient for inputs with different subtracted pulses and for an input with a constant subtracted value The un-pulsed and pulsed steering wheel angle input and vehicle trajectory Rollover coefficient for different pulse forms and for the constant added value Self-aligning moment in ADAMS and Maple for a symmetric pulse with a frequency of 1 Hz and 4 Hz Self-aligning moment in ADAMS and Maple for a non-symmetric pulse with a frequency of 1 Hz and 4 Hz Gear-train assembly design multiple-bar mechanisms adjustable-amplitude mechanisms Steering system Torque versus steering wheel acceleration Torque and power versus the frequency for different pulse forms and peak-time values Torque and power versus the amplitude for the symmetric pulse for different frequencies Torque and power versus the amplitude for the C continuous non-symmetric pulse for different frequencies Torque and power versus the amplitude for the C 1 continuous non-symmetric pulse for different frequencies A.1 Rollover coefficient for a pulse with a frequency of 8 Hz and an amplitude of 12 degrees between t = C.1 Torque (Nm) versus speed (rpm) iii

5 D.1 Tooth parts iv

6 List of Tables 3.1 Parameters of the used demo vehicle model maximum frequency for each ratio based on the maximum rotational speed and torque Planetary gear-set and worm gear data B.1 Values a, b and c for different frequencies used for the non-symmetric pulse. 44 C.1 Technical data of the motor v

7 Glossary NHTSA SUV ASC DYC HC ARS PASC δ s δ i s L R a y η g A f t a b c R F z,r F z,l m s M T r h e a y,s v y u r National Highway Traffic Safety Administration Sports Utility Vehicle Active Steering Control Direct Yaw Moment Control Hybrid Control Active Rear wheel Steering Pulsed Active Steering Control steering wheel angle steering angle of the front wheels steer ratio wheelbase corner radius lateral acceleration understeer coefficient gravity constant pulse amplitude pulse frequency time pulse peak-time value falling slope value rising slope value rollover coefficient vertical tire load on the vehicle s right hand side vertical tire load on the vehicle s left hand side vehicle sprung mass total vehicle mass track width height of CG above ground distance between CG and roll axis lateral acceleration of sprung mass lateral acceleration of total mass longitudinal velocity yaw rate vi

8 φ T M z R i ω i z T i N P D T s I eq b eq k eq δ s δ s δ s M z r T ps P s R δ sun A f t ω sun ω sun,max T sun T sun,max roll acceleration of vehicle vibration time self-aligning moment radius of gear i rotational speed of gear i ratio between ring-gear and sun-gear torque on gear i number of teeth on gear diametral pitch of gear pitch diameter of gear torque on steering wheel equivalent inertia of steering system equivalent damping of steering system equivalent stiffness of steering system acceleration of steering wheel angular velocity of steering wheel angle of steering wheel self-aligning moment scale factor torque delivered by power steering system power on steering wheel ratio between worm-gear and ring-gear angle of sun-gear pulse amplitude pulse frequency time rotational speed of sun-gear maximum rotational speed of sun-gear torque on sun-gear maximum torque on sun-gear vii

9 Contents 1 Introduction Research goals Report Overview Literature Review 3 3 Pulsed Active Steering effects ADAMS simulations Vehicle dynamics with pulse subtraction Symmetric pulse input Non-symmetric pulse input Optimal subtraction method Vehicle dynamics with pulse addition Discussion Results validation DFP and Maple simulations Simulation results Discussion Pulse actuation system Gear-train assembly Pulse actuator Power/Torque calculation Worm-gear design Discussion Conclusions and recommendations Conclusions Recommendations Bibliography 41 A Pulse during an extended time 43 B Non-symmetric pulse values 44 C Motor characteristics 45 viii

10 D Calculation ring gear thickness 46 ix

11 Chapter 1 Introduction Statistics from the National Highway Traffic Safety Administration (NHTSA) show that 9,362 of the total of 3,521 traffic fatalities in the United States in 26 are due to rollover of the vehicle. Sport Utility Vehicles (SUVs) had the highest rollover involvement rate of any vehicle type in fatal crashes: 35 % for SUVs, 28% for pickups, 17 % for vans and 17 % for passenger cars. In ,318 fatalities occurred due to rollover of the vehicle, so the amount of rollover crashes 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 and ride comfort of the vehicle. Much research has already been performed in the vehicle motion control area to control the vehicle (dynamics). Four main control techniques have been studied widely. One technique focusses 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 Yaw Moment Control, DYC), one focusses on controlling both the front wheels and the braking force distribution (Hybrid Control, HC) and one technique focusses on controlling the steering angle 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 able to prevent vehicle rollover. He claims to have shown that none of these systems are efficient enough to decrease the rollover of the vehicle. Therefore, he has proposed the Pulsed Active Steering Control System. He states that this new system is able to lower the chance of vehicle rollover efficiently. However, different pulse forms, frequencies and amplitudes show different effects on the vehicle rollover and on the vehicle trajectory. Since rollover crashes can occur for example by avoiding an obstacle on the road, it is also important that the vehicle trajectory is not changed too much due to the anti-rollover system. The exact effects of the Pulsed Active Steering Control system on both the rollover as the vehicle trajectory have not been studied into detail by Kuo, so more investigation needs to be performed. To study mechanical effects of this Pulsed Active Steering System on the total steering system and to validate the simulation results experimentally, a test setup needs to be build as well. 1

12 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 trajectory and rollover for different driving maneuvers and circumstances. Chapter 3 gives information about the simulations performed to investigate the effect of different pulse forms, amplitudes and frequencies on the vehicle trajectory and rollover and shows the simulation results. The effects are investigated for a steering wheel input with a subtracted pulse and with an added pulse. The effects are compared to a steering wheel input with a constant subtracted or added value to see if the Pulsed Active Steering System is able to reduce the vehicle rollover more than the Active Steering System. Chapter 4 describes the validation of the simulation results described in Chapter 3. This is done by comparing the self-aligning moment obtained by the software program ADAMS with 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 assembly and 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 an optimized maximum frequency to the steering wheel input. For this the maximum torque and 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 report and discusses some possible future research to improve the Pulsed Active Steering System. 2

13 Chapter 2 Literature Review This section will give 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 trajectory and rollover for different driving maneuvers and circumstances. Different controllers have been designed for the same control technique and some are therefore also addressed in this section. Advantages of active steering for vehicle dynamics control Ackermann et al. [2] have discussed the potential of DYC and ASC for yaw disturbance attenuation in terms of physical limits. They state that ASC only requires one fourth of the front wheel tire force compared to DYC. They also claim that ASC is better to generate a corrective torque to compensate torques caused by asymmetric braking and still have braking force left for acceleration. Asymmetric braking can arise due to a so called µ-split braking situation; the contact surface for wheels on the right hand side of the vehicle is dry, while the contact surface for wheels on the left hand side of the wheels is icy. If the DYC system generates 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 focusses on the control of the yaw motion and consists of decoupling of the vehicle s yaw and lateral motion as first presented in [3]. They state that this concept separates two basic tasks which have been the responsibility of the driver up until now: path following and disturbance attenuation. The first task is still left to the driver, but the disturbance attenuation can be controlled by the active steering system, making driving a vehicle easier and safer. Simulations have shown excellent disturbance rejection in µ-split braking and side-wind maneuvers. The second concept focusses on vehicle rollover avoidance by active steering and braking as first proposed in [4]. The presented controller consists of 3 feedback loops: emergency steering control, emergency braking control and continuous operation steering control. If the rollover coefficient (defined in [5]) reaches e.g. a value of.9 (rollover limit: R = 1) due to a high driver s 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 same time vehicle deceleration occurs through braking and the chance of vehicle rollover is further reduced. By controlling the braking pressure the vehicle trajectory is maintained according to the driver s steering command. The continuous operation steering control is added to improve the vehicle s roll-damping and roll-disturbance attenuation. Simulations have shown that this 3

14 control setup is able to prevent rollover and is able to maintain practically the same vehicle trajectory 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, the system is designed such that the the performance of the actual vehicle model follows that of an ideal vehicle model. The actual vehicle model is described as a bicycle model including direct yaw moment input. The desired vehicle model has been derived by the control law of ARS in which the rear wheels are steered such that the vehicle body sideslip is zero. The proposed model-matching controller consists of the desired model and a feed-forward and feedback compensator. The feed-forward compensator decides the control inputs; the front wheel steering angle and the direct yaw moment generated from braking forces. The feedback compensator 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 of the vehicle is improved by this system compared to these motions when only a DYC system is used. The simulations also show that the system has a robust performance to make the actual 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 references therein) and evaluated this system by simulating different driving events. The inputs of the vehicle model are the steering angle set by the driver and a side wind force. A steering angle contribution is derived using the yaw rate and the steering wheel angle and this contribution is 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 force disturbances,that the controlled vehicle is harder to make unstable than the uncontrolled vehicle 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 prevent vehicle rollover. A nonlinear 4 degree of freedom vehicle yaw/roll model as well as a complex nonlinear tire model have been derived and used for these simulations. He claims that this new model represents the real-world vehicle to a good degree of accuracy. Using an ASC system, the results show that the rollover coefficient is reduced to a small proportion of the original magnitude, but the system is not able to fully reduce the vehicle rollover below a certain threshold when the rollover is too high due to an extreme drivers steering input. The DYC system also seems to be unable to prevent rollover at high vehicle speeds and extreme driver steering inputs. This is because the high braking forces needed to decrease the rollover result in a significant shift in vertical tire load to one of the front tires, causing abnormal tire lateral forces. This results in vehicle instability. The HCS shows better results compared to the other two controllers, but since it also includes the differential braking mechanism, it is sensitive to vertical tire load shift and would therefore also fail to prevent rollover. 4

15 Therefore, Kuo has designed and tested a slightly new vehicle rollover control strategy, the Pulsed Active Steering Control (PASC) system. The difference between the Active Steering Control system and the Pulsed Active Steering Control system is that, instead of a constant value, a pulse with a certain amplitude and frequency is added or subtracted to the steering wheel input given by the driver. The only input of the designed controller is the steering wheel input given by the driver. By calculating different variables the rollover coefficient is calculated. If this exceeds a designated threshold a pulse is subtracted from the original driver steering input. Simulations show that using a symmetric pulse results in a rollover with sudden bumps higher than the rollover obtained for the un-controlled vehicle. Using a non-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. The non-symmetric pulse used consists of a smooth curve with a sharp, gradually decreasing slope combined with a smooth, gradually increasing slope. Compared to a symmetric pulse and a square pulse, this pulse shows a smaller reduction of the rollover coefficient in its total amount, 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 able to prevent rollover. However, it is also visible that the controlled vehicle trajectory is different from the uncontrolled trajectory. Simulating at different frequencies shows that if the frequency is either too high or too low, the efficiency of the controller is reduced. It is also visible that different pulse frequencies result in different vehicle trajectories. Overall, the simulations show that the pulse amplitude, the pulse frequency and the threshold of the rollover coefficient to trigger the controller are the three important control variables essential for a well-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 yaw dynamics. The results of these controllers have been compared with a simulation model based on an enhanced bicycle model. In this model the tire relaxation length and suspension steering compliance have been taken into account. The first controller, the yaw rate feedback controller, consists of a reference model and a rear wheel steering controller. The controller is designed to minimize the yaw rate overshoot, since this overshoot is undesirable and leads to an increased workload for the driver. The reference model provides the reference yaw rate and is compared to the actual vehicle yaw rate. The difference is fed back to the steering controller. Simulations show that this active rear wheel steering control system is able to suppress the undesired yaw rate overshoot. The disadvantage of this controller is that on a real vehicle an accurate yaw rate signal is 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 the yaw velocity oscillation is not related to the frequency of the original yaw oscillation. This means that it is not necessary to apply counter steering at the rear wheels depending on the yaw velocity oscillation. Therefore, a feed-forward rear wheel steering controller has been designed. Using the relation between the step response of the rear wheel steering angle and the front steering angle, as found for the feedback controller, a transfer function is proposed to 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 are necessary. Simulations show that this controller is able to eliminate the yaw velocity overshoot 5

16 and oscillations without the need of an accurate yaw rate sensor. The performances of this system are almost the same as for the feedback controller. 6

17 Chapter 3 Pulsed Active Steering effects To investigate the effect of a Pulsed Active Steering Control system (PASC) on the vehicle trajectory and rollover, simulations are performed using a steering wheel input with different subtracted or added pulse forms, frequencies and amplitudes. The simulations are performed with the mechanical system simulation software program MSC.ADAMS. The first goal of the PASC is to decrease the vehicle rollover as much as possible without changing the vehicle trajectory too much. This can be done by decreasing the driver s steering input. Therefore, the effects of a steering wheel input with a subtracted pulse is investigated first. The second goal of the PASC is to use the system for track following. The deviation from a desired trajectory due to understeer for example can be decreased by increasing the driver s steering input. Therefore, the effects of a steering wheel input with an added pulse is investigated second. Some of the results are compared to a steering wheel input with a constant (un-pulsed) subtracted or added value. This is done to investigate if the PASC system works better than the ASC. 3.1 ADAMS simulations The software program MSC.ADAMS makes it possible to simulate the full-motion behavior of a complex mechanical system and to analyze multiple design variations or motion inputs in a fast way. All the simulations are made using the non-linear demo vehicle model provided by the program. The tire-model used is Pacejka 22 consisting of the Magic Formula for both longitudinal and lateral tire forces, the transient response to friction changes and the slip dependent relaxation effect. Parameters of the demo vehicle model are shown in Table 3.1. The vehicle motions are defined according to the SAE sign convention, as indicated in Figure 3.1. The steering wheel angle (δ s ) given by the driver is chosen as input for all simulations. The resulting steering angle of the front wheels (δ) can be found by dividing the steering wheel angle by the steer ratio (i s ). The steer ratio of the vehicle model used can be found by simulating steady-state cornering. For steady-state cornering the steering angle of the front wheels can be found by the equation: δ = L R + a y g η (3.1) 7

18 Definition Symbol Unit Value Total vehicle mass m kg 153 Vehicle sprung mass m s kg 143 Wheel base L m 2.56 Track width front w f m 1.52 Track width rear w r m 1.59 Distance from center of gravity to front axle L f m 1.48 Distance from center of gravity to rear axle L m m 1.77 Height of center of gravity above ground h m.432 Spring stiffness K N/m 1.25e5 Vehicle moment of inertia w.r.t. x-axis I xx kgm Vehicle moment of inertia w.r.t. y-axis I yy kgm Vehicle moment of inertia w.r.t. z-axis I zz kgm Table 3.1: Parameters of the used demo vehicle model Figure 3.1: Vehicle motions defined according to the SAE convention 8

19 With L the wheelbase, R the corner radius, a y the lateral acceleration and η the understeer coefficient of the vehicle and g the gravity constant. The understeer coefficient determines whether the steering angle needs to be changed to remain a certain constant radius R if the forward speed of the vehicle is increased. For a neutral vehicle the understeer coefficient is zero and the steering angle can remain the same, for a understeered vehicle the understeer coefficient is higher than zero and the steering angle needs to be increased and for an oversteered vehicle the understeer coefficient is lower than zero and the steering angle needs to be decreased. Simulating steady-state cornering at different vehicle speeds shows that the vehicle model used in ADAMS has understeer. The exact understeer coefficient has not been determined, since it is not important for the investigation performed in this report. To calculate the steer ratio the steady-state cornering needs to be simulated at a low vehicle speed. At low speeds the lateral acceleration of the vehicle is very low and the effect of the understeer coefficient can therefore be neglected. The equation of the steer ratio than becomes: i s = δ s δ = δ s L R + ay g η = δ sr L Using a steering wheel input of 3 degrees for the steady-state cornering simulation a resulting corner radius of 11.5 meters is found. These values result in a steer ratio of 23.5 for the demo vehicle model. The driving maneuver and the different pulse forms used for the simulations and a way to investigate the vehicle rollover are described next. Driving maneuver It is expected that the influence of different pulse forms, frequencies and amplitudes on the vehicle 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. A representation of the simulation input for this maneuver can be found in Figure 3.6. As can be seen, after one second the steering input gradually increases to a maximum within one second and stays here from the 2 nd to the 5 th second. From the 5 th to the 6 th second the steering input gradually decreases back to degrees. Pulse forms (3.2) The effect of two different pulse forms are investigated: a symmetric pulse and a nonsymmetric pulse. The symmetric pulse is given by the following equation: y(t,f) = A(1 cos(2πft)) (3.3) with A the pulse amplitude, f the pulse frequency and t the time. A representation of this symmetric pulse is shown in Figure 3.3. The non-symmetric pulse is the one recommended by Kuo. According to his findings, this special pulse form is more useful in reducing the rollover coefficient than the symmetric pulse is. The pulse form consist of a sharp, gradually decreasing slope (given by y 1 ) combined with a smooth, gradually increasing slope (given by y 2 ): y 1 (t) = 2Ae (t a)2 b for t a (3.4) y 2 (t) = 2Ae (t a)2 c for a t (3.5) 9

20 Steering wheel input vs time J turn maneuver input steering wheel angle δ s [deg] time [s] 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 where the pulse reaches its peak value and the values b and c give the shape of the falling and rising slope, respectively. A representation of the shape of this non-symmetric pulse with a =.25, b =.5 and c =.45 is also shown in Figure Amplitude symmetric pulse non symmetric pulse.4 y.6 b c.8 a Time [s] 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, all the 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 software program. The effects on the vehicle rollover is investigated by calculating the rollover coefficient, which is a measure for the rollover risk [8]. The coefficient is given by the following equation: R o = F z,r F z,l = 2m s {((h e) + ecos ϕ) a y,s + esin ϕ} (3.6) F z,r + F z,l MT r g With F z,r and F z,l the vertical tire load on the right hand side and the left hand side respectively, m s the vehicle sprung mass, M the total vehicle mass, T the track width, h the 1

21 height of the center of gravity above the ground, e the distance between the center of gravity and the roll axis, ϕ the roll angle and g the gravity constant. a y,s is the lateral acceleration of the sprung mass and is given by the equation: a y,s = v y + ur e ϕ (3.7) with v y the lateral acceleration of the total mass, u the longitudinal velocity, r the yaw rate and ϕ the roll acceleration of the vehicle. The non-linear vehicle model and all the parameters and variables used are shown in figures 3.4 and 3.5. The vehicle is about to rollover if the tire loads on one side of the vehicle become zero. At that moment the absolute value of the rollover 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 the steering wheel input are given, followed by the effects of adding these pulses to the steering wheel input. 11

22 3.2 Vehicle dynamics with pulse subtraction The primary goal of the PASC is to lower the rollover coefficient, but the vehicle trajectory intended by the driver can not be changed too much by the anti-rollover system if an obstacle on the road needs to be avoided. Therefore, the results of the simulations made for a steering wheel input with a subtracted pulse are compared to an un-pulsed input, since this gives the drivers intended uncontrolled vehicle trajectory. The steering wheel angle used for the J-turn maneuver for the un-pulsed input and for the input with a subtracted pulse can be found in Figure 3.6. As can be seen, the maximum angle of the steering wheel input is chosen to be 32 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 4 km/h. One might be expecting 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 oscillation and the effect of pulse subtraction is therefore less easy to investigate. The used input has shown to work well for the investigation. steering wheel angle δ s [deg] Steering wheel input vs time Pulse amplitude pulsed input un pulsed input time [s] Figure 3.6: The pulsed and un-pulsed steering wheel input δ s for the J-turn maneuver First the results of subtracting a symmetric pulse will be given in section 3.2.1, followed by subtracting a non-symmetric pulse in section In section the rollover coefficient of both pulses will be compared to a steering wheel input with a subtracted constant value Symmetric pulse input The effects of different frequencies using a symmetric pulse is investigated for two pulse amplitudes. In [8] the ratio between the maximum steering angle of the front wheels of the uncontrolled vehicle and the pulse amplitude for the J-turn maneuver is around 5:2. For ease of comparison this ratio is used for the first simulation, resulting in a pulse amplitude of 12 degrees. For the second simulation the amplitude is decreased to an arbitrary 8 degrees. The resulting vehicle trajectory for both amplitudes at different frequencies is shown in figures 3.7 (a) and (b), respectively. The un-pulsed vehicle trajectory is also shown in both figures. The following conclusions can be drawn from these figures: Increasing the frequency from 1 to 4 Hz results in a larger path deviation with respect to the un-pulsed input, independent of the pulse amplitude. 12

23 A high frequency of 8 Hz results in a smaller path deviation compared to the 4 Hz frequency. A higher pulse amplitude results in a larger path deviation with respect to the un-pulsed input for all frequencies. The difference in the vehicle trajectory between the frequencies depends on the pulse amplitude. Vehicle trajectory Vehicle trajectory y [m] 4 y [m] 4 5 unpulsed 6 sym. pulse 1 Hz sym. pulse 2 Hz 7 sym. pulse 4 Hz sym. pulse 8 Hz x [m] 5 unpulsed 6 sym. pulse 1 Hz sym. pulse 2 Hz 7 sym. pulse 4 Hz sym. pulse 8 Hz x [m] (a) Pulse amplitude 12 degrees (b) Pulse amplitude 8 degrees Figure 3.7: Vehicle trajectory for a pulse with an amplitude of 12 and 8 degrees for different frequencies It is clear that the vehicle trajectory depends on the frequency, but most of all on the pulse amplitude. This is due to the fact that the pulse amplitude is the biggest factor determining the overall average input, since the pulse amplitude determines the amount of degrees subtracted from the steering wheel input. Note that the vehicle trajectory for pulses with an amplitude of 12 degrees and with a frequency of 1 and 2 Hz are almost the same, but this is not the case for the pulse with the lower amplitude. Hence, it might be concluded that high amplitude pulses with a small frequency have almost the same effect on the vehicle trajectory. More investigation is necessary to confirm this conclusion. The rollover coefficient for both simulations are shown in figures 3.8 (a) and (b), respectively. The following conclusions can be drawn from these figures if one looks at the results during the time the pulse is being subtracted: Pulses with a frequency of 1 and 2 Hz result in a higher rollover coefficient compared to 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 to the 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 of 4 and 8 Hz. 13

24 The difference in the rollover coefficient between the frequencies depends on the pulse amplitude. Rollover coefficient vs time Rollover coefficient vs time Rollover coefficient unpulsed sym. pulse 1 Hz sym. pulse 2 Hz sym. pulse 4 Hz sym. pulse 8 Hz Rollover coefficient unpulsed sym. pulse 1 Hz sym. pulse 2 Hz sym. pulse 4 Hz sym. pulse 8 Hz Time [s] (a) Pulse amplitude 12 degrees Time [s] (b) Pulse amplitude 8 degrees Figure 3.8: Rollover coefficients for a pulse with an amplitude of 12 and 8 degrees for different frequencies A possible explanation of the fact that low frequencies result in a high rollover coefficient is that these frequencies are close to the eigenfrequency of the suspended vehicle body. The fact that a higher pulse amplitude results in a lower rollover coefficient is due to the fact that for high amplitudes more degrees are subtracted from the steering wheel input. This results in a lower overall average steering wheel input and therefore in a less extreme J-turn maneuver. 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 rollover coefficient compared to a pulse with a frequency of 4 Hz. The general effects of different frequencies on the vehicle trajectory and rollover coefficient do not depend on the amplitude of the pulse. Note that the maximum rollover coefficient does not reach the critical value of 1. This is due to the input used for the simulations and due to the fact that the demo vehicle used in the software program has a low center of gravity, which makes rollover more difficult. It is also important to note that the rollover coefficient is only decreased for frequencies above the 4 Hz during the time the pulse is being subtracted. To make sure that the rollover coefficient stays 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 control system. 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 found in Appendix A. From this result it can be concluded that an increased pulse subtraction 14

25 time results in a lower rollover coefficient compared to the un-pulsed input during the total maneuver time. The differences with the simulation with the shorter pulse-subtraction-time are very small during this interval. Hence, it appears that the conclusions drawn based on the previous results do not depend on the time the pulse is subtracted Non-symmetric pulse input Studying the effect of the non-symmetric pulse as described in section 3.1 is done by first researching the effect of different frequencies and secondly, by researching the effect of an increased pulse peak-time value a for a constant frequency. This last research is performed since it is expected that the peak-time value also has a big effect on the vehicle trajectory and rollover. The amplitude for all simulations is chosen to be 12 degrees (also used for the symmetric pulse), since at this amplitude the rollover coefficient is being reduced the most. It can be expected that the conclusions drawn from these simulation also hold for smaller or bigger amplitudes. Frequency modulation To investigate the effect of different frequencies, the peak-time value a (as function of the vibration time T) must be kept constant. For this investigation the value is chosen to be 1 4 of the vibration time. The begin and end points of each pulse need to be the same and almost equal to. Therefore, the values b and c as used in (3.4) and (3.5) need to be determined by the following equation: f 1,i (t = ) = f 2,i (t = T) = constant for i = 1,2,4,8 Hz (3.8) Note that this equation only holds for one pulse starting at time t =. For a pulse with a frequency of 1 Hz, the value b is chosen to be.5. Using this combination of values for a and b the constant value in equation 2.5 is found (3.7267e 6 ) and the value c can now also be determined using the same equation. Using (3.4), (3.5) and (3.8), the values b and c as function of the frequency can be calculated by: b = c = a2 = 12.5 (T a)2 a 2 b = 1 2f 2 (3.9) 9 2f 2 (3.1) Using these equations a combination of values a,b and c is found for each frequency (see Appendix B). These combinations are used for the simulations. The resulting vehicle trajectories at each frequency are shown in Figure 3.9 (a). The conclusions 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 the non-symmetric pulse are shown in Figure 3.9 (b). It can be seen that the non-symmetric pulse results in a smaller path deviation. This is as expected, since the area above the pulse is lower for the non-symmetric pulse. This means that less is subtracted from the steering wheel input, causing a higher average steering input and a lower path deviation. 15

26 Vehicle trajectory Vehicle trajectory y [m] 4 y [m] unpulsed non sym. pulse 1 Hz non sym. pulse 2 Hz non sym. pulse 4 Hz non sym. pulse 8 Hz x [m] (a) Vehicle trajectory for a non-symmetric pulse with different frequencies unpulsed average symmetric pulse average non symmetric pulse x [m] (b) Average vehicle trajectory for the symmetric and non-symmetric pulse Figure 3.9: Vehicle trajectories The rollover coefficient at each frequency can be found in Figure 3.1 (a). Figure 3.1 (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 symmetric pulse, but this specific non-symmetric pulse seems to be unable to decrease the rollover with respect to the uncontrolled input. Rollover coefficient Rollover coefficient versus time unpulsed non sym. pulse 1 Hz non sym. pulse 2 Hz non sym. pulse 4 Hz non sym. pulse 8 Hz Rollover coefficient Rollover coefficient versus time unpulsed non sym. pulse 1 Hz non sym. pulse 2 Hz non sym. pulse 4 Hz non sym. pulse 8 Hz Time [s] (a) Unzoomed Time [s] (b) Zoomed Figure 3.1: Rollover coefficient for the non-symmetric pulse with an amplitude of 12 degrees Pulse peak-time modulation To determine the effect of the peak-time value a, one simulation is made using a pulse with a value of a = 3 4T. This means that the pulse now consists of a smooth, gradually decreasing slope combined with a sharp, gradually increasing slope. The simulation is performed for a pulse with an amplitude of 12 degrees and a frequency of 8 Hz, since this high frequency results in the lowest rollover coefficient. Note that changing the value a also implies a change in the values of b and c. 16

27 The vehicle trajectory and the rollover coefficient for this simulation is shown in figures 3.11 (a) and (b), respectively. The vehicle trajectory and the rollover coefficient for the symmetric pulse and for the non-symmetric pulse with the old value of a = 1 4T are added for comparison. As can be seen, a pulse with a high peak-time value results in a slightly larger path deviation with respect to a pulse with a low peak-time value. However, the rollover coefficient is significantly lower for the pulse with a high peak-time value. Hence, it can be concluded that a pulse with a high a-value seems to have good potential to decrease the rollover coefficient combined with a small path deviation. However, the symmetric pulse shows the lowest rollover coefficient, although it also shows a larger path deviation. y [m] Vehicle trajectory Rollover coefficient Rollover coefficient versus time unpulsed non sym. pulse, a = 1/4 T non sym. pulse, a = 3/4 T sym. pulse x [m] (a) Vehicle trajectory Time [s] (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 investigated separately. The next step is to combine the two and to investigate which pulse form can best be subtracted from the steering wheel input to decrease the rollover coefficient the most. This investigation is performed next Optimal subtraction method For a good comparison between the rollover coefficient of each pulse form the vehicle trajectory needs to be the same for all. The same vehicle trajectory can be obtained by modifying the amplitude of each pulse. The investigation is only performed for pulses with a frequency of 8 Hz, since this high frequency decreases the rollover coefficient the most. It is found that decreasing the amplitude of the symmetric pulse from 12 to 76 degrees gives the same vehicle trajectory as the non-symmetric pulse with an amplitude of 12 degrees and a peak-time value of 3 4T (see section 3.2.2). The simulation results are also compared to the rollover coefficient obtained from a steering wheel input with a constant (un-pulsed) subtracted value. This makes it possible to investigate if the PASC system is better able to decrease the vehicle rollover than the ASC system. Before a proper comparison can be ensured it needs be noted that one of the reasons the non-symmetric pulse used in section results in a relatively high rollover coefficient can be that the falling and rising slope of the pulse are too sharp. A second reason can be that the non-symmetric pulse used is C continuous. ADAMS might not be able to work well with a C continuous pulse, probably due to interpolation problems. A C continuous 17

28 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 C 1 continuous non-symmetric pulse with a slightly less sharp falling and rising slope. This new pulse is given by the following equation: f(t,t) = A 2 (1 cos(eq(t mod(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(2π + 1) T n (3.12) n =.335 ( b +.46) (3.13) a The peak-time value is given by the factor b a. A representation of this pulse with a peaktime value of 3 4 T can be found in Figure The C continuous non-symmetric pulse with the same peak-time value and the symmetric pulse are also shown for comparison y symmetric pulse C continous non sym. pulse C 1 continous non sym. pulse time Figure 3.12: Representation of the different pulse forms For the simulation the peak-time value of this new pulse form is chosen at 3 4T, since this peak-time value results in the lowest rollover coefficient for the C continuous non-symmetric pulse (see section ). The amplitude of the pulse is chosen such that the vehicle trajectory is the same as the inputs with the two other subtracted pulse forms and the same as the input with the constant subtracted value. The resulting rollover coefficients for the different inputs are given in Figure As can be seen in this figure, the new non-symmetric pulse results in a significant decrease in the rollover coefficient with respect to the other non-symmetric pulse and even shows a lower coefficient than the symmetric pulse. So the new non-symmetric pulse studied so far has the highest potential to decrease the rollover coefficient compared to other pulses. However, the lowest 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 subtracted value. Increasing the peak-time value of the non-symmetric pulses or increasing the pulse 18

29 Rollover coefficient Rollover coefficient versus time unpulsed non sym. pulse, a = 3/4 T sym. pulse, amp 76 new nsp, amp 72 constant subtraction Time [s] Figure 3.13: Rollover coefficient for inputs with different subtracted pulses and for an input with a constant subtracted value frequency even more than 8 Hz can possibly result in a decreased rollover coefficient, but it will always be higher than the input with the constant subtracted value. This means that it is best to subtract a constant value instead of a pulse from the steering wheel input to decrease the 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 investigated next. 3.3 Vehicle dynamics with pulse addition Due to understeer or wind disturbance for example, the vehicle can deviate from a desired trajectory. In section 3.2 it is found that modifying the amplitude of each different subtracted pulse results in a specific vehicle trajectory. This means that for a steering wheel input with an 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 furthermore that subtracting different pulse forms with a certain frequency results in a lower rollover coefficient compared to the uncontrolled input. So, adding the pulse will consequently result in a higher rollover coefficient. Two questions arise from above observations: First, which pulse form can best be used for track following without increasing the rollover coefficient too much and second, if adding a pulse to the steering wheel is better than adding a constant (un-pulsed) value to the steering wheel input. To investigate these questions the effect of adding different pulse forms on the rollover coefficient 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 pulse and therefore only the different pulses at 8 Hz are being compared. The simulated driving maneuver is again the J-turn maneuver. A representation of the vehicle steering wheel input used for this maneuver can be found in Figure 3.14 (a). The vehicle speed is chosen to be 4 km/h, as is used for all earlier performed simulations. As already noted, the vehicle model used in ADAMS has understeer and therefore the un-pulsed trajectory is not the desired 19

30 trajectory. The desired vehicle trajectory is determined at a speed of 3.6 km/h, since at low vehicle speeds the influence of the understeer coefficient on the vehicle trajectory can be neglected. The maximum uncontrolled steering wheel input is chosen to be 12 degrees. The desired vehicle trajectory and the un-pulsed vehicle trajectory are shown in Figure 3.14 (b). steering wheel angle δ s [deg] Steering wheel input vs time pulsed input un pulsed input Pulse amplitude y [m] desired trajectory un pulsed trajectory Vehicle trajectory time [s] (a) Steering wheel angle input x [m] (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 wheel input with the added pulse gives the desired trajectory. It is found that the amplitude of the symmetric pulse needs to be 25 degrees, the amplitude of the C continuous non-symmetric pulse 51 degrees and the amplitude of the C 1 continuous non-symmetric pulse needs to be 23 degrees to get the desired trajectory. So using these pulses with these amplitudes results in the black line visible in figure 3.14 (b). Note that the peak-time value is 3 4T for both non-symmetric pulses. The rollover coefficient for the different pulse forms and for the input with a constant added value are shown in Figure Rollover coefficient Rollover coefficient versus time unpulsed C cont. non sym. pulse C 1 cont. non sym. pulse symmetric pulse averaged un pulsed Time [s] 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 the symmetric pulse, closely followed by the C 1 continuous non-symmetric pulse. The steering wheel input with a constant added value results in the lowest rollover coefficient. The rollover 2

31 coefficient obtained from the steering wheel input with the different added pulses oscillates again around the steering wheel input with the constant added value. So changing the peaktime of the C 1 continuous non-symmetric pulse or increasing the frequency will result in a different rollover coefficient (as can also be seen in section 3.2.2), but the rollover coefficient will 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, the pulse can not always be added. If, due to a certain input, the rollover coefficient reaches a value 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 input with a constant added value will be able to decrease the path deviation the most, since it increases the rollover coefficient the least. 3.4 Discussion All the simulation results lead to the conclusion that the PASC system is able to decrease the rollover coefficient by subtracting a pulse and is able to decrease or even delete a path deviation by adding the pulse to the steering wheel input. A C 1 continuous non-symmetric pulse with a peak-time value of ( 3 )(4)T has the best potential to decrease the rollover coefficient compared to 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 to result in the best combination of rollover coefficient and vehicle trajectory. However, the best input is found to be the one with a subtracted or added constant value, since this results in the lowest rollover coefficient. This means that the ASC system works better than the PASC system. 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 research can 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 goal of the PASC system is to decrease the vehicle rollover of especially SUV, further research can be 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 pulse results in a rollover coefficient with some bumps higher than the uncontrolled vehicle rollover coefficient. The bumps found can be due to the fact that the symmetric pulse used has a C continuity at the beginning and end of each pulse. He has shown that the C continuous non-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 decrease the rollover coefficient below the uncontrolled vehicle rollover coefficient. This can be due too a too sharp falling and rising slope of the pulse or due too interpolation problems in the ADAMS software. More research can be performed upon this subject. Kuo also states that the overal reduction in its total amount is smaller for the C continuous non-symmetric pulse 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 to reduce the rollover coefficient as much as the new non-symmetric pulse proposed or even the symmetric pulse. Since different pulse forms give different results, it might be possible that there is a certain form which is able to decrease the rollover coefficient even more than the different forms used 21

32 in this report. It might also be that there is a certain pulse form which is able to decrease the rollover coefficient even more than subtracting a constant value. However, this is not expected since the rollover coefficient of all the pulses oscillate around the rollover coefficient given 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 system will result in excessive wear of the tyres. It is also expected that the system has a negative effect on the ride comfort, since rolling of the vehicle with a certain frequency can be annoying for 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

33 Chapter 4 Results validation To check whether the results obtained with the software program ADAMS as shown in Chapter 3 are acceptable, the simulation results are validated by comparing simulation results from ADAMS with simulation results obtained using the software program Maple in combination with DynaFlexPro (DFP). First information about the new software programs and about the performed simulations is given, followed by the simulation results. At the end a conclusion based upon these results is given. 4.1 DFP and Maple simulations The sofware program Maple is able to compute and manipulate symbolic expressions. The symbolic expressions, the kinematics and dynamic equations of a system, can be automatically generated by the program DFP wherein the studied system model is built. The model has fourteen degrees of freedom: six for the chassis (three translational and three rotational), one for the spin of each tire and one for the vertical prismatic joint of each suspension link. The input 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 for a SUV. For the design it needs to be known what the maximum amount of wheel angle of the front wheels is before a SUV starts to rollover. As already noted, for the simulations performed 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 the Chevrolet Equinox, which is a SUV. Therefore, DFP in combination with Maple is first used to determine this maximum amount of wheel angle input. This information will also be used for the validation of the ADAMS vehicle model. For this first investigation the J-turn maneuver as described in section 2.1 is simulated at a vehicle velocity of 8 km/h. This velocity is chosen since it is more likely that the vehicle rolls over at this high velocity compared to the relatively low velocity of 4 km/h used for the simulations in Chapter 3. It is found that an input on the front wheels above 3 degrees results in rollover of the Equinox model. This maximum steer angle of the front wheels is chosen as input for the simulations in Maple. Using the steer ratio of 23.5 found in section 2.1, the maximum steering wheel input for the ADAMS simulations becomes 7.5 degrees. Simulations are performed using a steering wheel input with a subtracted symmetric pulse 23

34 and a subtracted C continuous non-symmetric pulse with a frequency of 1 and 4 Hz. One might expect a frequency of 8 Hz, because this results in the lowest rollover coefficient, but the input for this simulation in Maple would be too extensive. The used frequencies are expected to work well for the validations and it is expected that the results found also hold for other frequencies. 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 amplitude of the pulse as also used for the simulations in Chapter 3. Using the steer ratio of 23.5 this results 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 (M z ). The self-aligning moment is caused by the lateral force of the tire produced by the slip angle. The force acts through a point behind the center of the wheel, the pneumatic trail, in a direction such that it attempts to re-align the tire. The total self-aligning moment of the front wheels can be calculated by adding the self-aligning moment of the left wheel with the self-aligning moment of the right wheel. For a good comparison the vehicle parameters used in the program ADAMS (as given in Table 2.1) are implemented in the vehicle model built in DFP. So the model used in DFP resembles 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 frequency of 1 Hz and 4 Hz given by both programs are presented in figures 4.1 (a) and (b), respectively. The self-aligning moment for the input with a subtracted non-symmetric pulse with a frequency of 1 Hz and 4 Hz are presented in figures 4.2 (a) and (b), respectively. As can be seen 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 acts counterclockwise and since a moment clockwise is taken as positive, the resulting self-aligning moment is negative. Furthermore it can be seen that the minimum of the self-aligning moment is lower for the DFP model than for the ADAMS model. This is due to the difference in the models. The most important difference between the models is the difference in suspension: in DFP the model used has a (simplified) vertical suspension, while ADAMS uses a McPherson suspension. This can also explain the fact that the self-aligning moment in the ADAMS 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-aligning moment can be directly related to the applied torque on the steering wheel. The applied torque on the steering wheel to turn the front wheels is information needed to design the pulse actuation system proposed in Chapter 4. However, it is found that the torque on the steering wheel depends on geometric parameters of the steering system [1]. These geometric variables are unknown and therefore it is not possible to relate the self-aligning moment directly to the torque on the steering wheel. The results obtained by both software programs show a distinctive difference, but these differences can be explained by the difference in the models used. The differences seem to be 24

35 2 Self aligning moment vs time ADAMS Maple 2 Self aligning moment vs time ADAMS Maple 2 2 M z [Nm] 4 M z [Nm] Time [s] Time [s] (a) 1 Hz (b) 4 Hz Figure 4.1: Self-aligning moment in ADAMS and Maple for a symmetric pulse with a frequency of 1 Hz and 4 Hz 2 Self aligning moment vs time ADAMS Maple 2 Self aligning moment vs time ADAMS Maple 2 2 M z [Nm] 4 M z [Nm] Time [s] Time [s] (a) 1 Hz (b) 4 Hz Figure 4.2: Self-aligning moment in ADAMS and Maple for a non-symmetric pulse with a frequency of 1 Hz and 4 Hz 25

36 consistent for different pulse forms and for different frequencies. From this it can be concluded that 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 the pulse actuation system for the setup. 26

37 Chapter 5 Pulse actuation system The results of the performed simulations given in chapter 2 show that the PASC system has good potential to lower the vehicle rollover. To study the mechanical effect of the PASC system on the total mechanical steering system and to perform experiments for the validation of the results found in Chapter 3 and 4 a test setup needs to be built. This setup will consists of a steering column, steering rack, a set of wheels and a pulse actuation system. This pulse actuation system adds or subtracts the pulse to the driver s steering wheel input and will be placed between the steering wheel column and the steering pinion/rack. The design of the pulse actuation system is described in this chapter. The pulse actuation system consists of a gear-train assembly 1 and a pulse actuator. The design 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 disadvantages a 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 pulse frequency can be added or subtracted to the driver s steering wheel input. For the design it is necessary to obtain the maximum torque and power needed to generate the pulse motions of 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 is 1:1 if no pulse is applied. The steering wheel input and output are co-linear, which means that the input and output 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 high as possible. 1 A first setup of this assembly has been designed by Alexander Berlin, a student at the University of Waterloo 27

38 Figure 5.1 (a) shows a 3-dimensional drawing and Figure 5.1 (b) shows the working scheme of the 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 4 are necessary to make the steering wheel input versus the output of the total system 1:1, if no pulse is applied. The planetary gear-set consists of the sun (gear 8), the ring (gear 7) and three planets (gears 6) connected to the carrier (gear 5). The carrier is directly connected to gear 4. The sun-gear is connected to the steer-rack and is the output of the assembly. The pulse will be applied on the ring gear by the pulse actuator. Details about the pulse actuator can 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: R 1 ω 1 = R 2 ω 2 (5.1) ω 2 = ω 3 (5.2) R 3 ω 3 = R 4 ω 4 (5.3) ω 4 = ω 5 (5.4) ω 8 = (z + 1)ω 5 zω 7 (5.5) R 7 = R 8 + 2R 6 (5.6) T 7 = zt 8 (5.7) T 5 = (z + 1)T 8 (5.8) With R i the radius of gear i, ω i the rotational speed of gear i, z the ratio between the ringgear and sun-gear (z = R 7 R 8 ) and T i the torque on gear i. When the ring gear is stationary (ω 7 = ), the input versus output (ω 1 : ω 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 = R 2R 4 R 1 R 3 (5.9) To make the steering wheel input and planetary gear-set input co-linear, the following equation has to hold as well: R 1 + R 2 = R 3 + R 4 (5.1) 28

39 The number of teeth of each gear can now be chosen such that all the above equations hold, but it needs to be taken into account that the gears of the gear train assembly need to be provided by the company Boston Gears [11]. The system also has to be as cheap as possible and as compact as possible and the gears need to be able to withstand the maximum applied torque and power. These last two constraints depend on the pressure angle, number of 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 normal to the tooth surface and the plane tangent to the pitch surface. The company supplies gears with pressure angles of 14.5 and 2. Gears with a higher pressure angle have a higher load carrying capacity, but gears with a lower pressure angle are better for extensive use, have less change in backlash and have a higher contact ratio and therefore a smoother and quieter operation [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 will result in a bigger total gear diameter and therefore violates the compact constraint. Taking a ratio of 1.5 and using the gears provided by the company, the smallest diameter of gears is found 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 5.1 hold. This results in 16 teeth for gear 1, 2 teeth for gear 2, 12 teeth for gear 3 and 24 teeth for gear 4. Diametral pitch The gear supplier provides gears not only with different numbers of teeth (N), but also with a a different diametral pitch (P). The diametral pitch is the number of teeth in the gear for each inch of pitch diameter. Both variables determine the pitch diameter (D) of a gear by the following equation: D = N P (5.11) The diametral pitch is an important factor determining the maximum torque and power that the gear can handle: the higher the diametral pitch, the lower the maximum torque and power that the gear can withstand. The maximum torque and power supplied to the gears depend on the motor driving the pulse actuator. The pulse actuator will be driven by a motor available at the University of Waterloo. This available motor is the Kollmorgen Seidel 6SM47L-3. The rated speed of this motor is 3 rpm, the rated torque at this rated speed is 2.2 Nm and the rated power is 69 W. More motor characteristics can be found in Appendix C. Using the rated power and the approximated horsepower and torque ratings provided by the gear supplier, it is found that the diametral pitch of the gears has to be 12 or less, otherwise the smallest gears (the planets on the planetary gear-set) will not be able to withstand the supplied power. Since a smaller diametral pitch results in an undesired bigger diameter of the gears a diametral pitch of 12 is chosen for now. The maximum torque and power that a gear can handle does not 29

40 only depend on the diametral pitch, but also on the rotational speed of the gear. At a lower speed the gear can handle a lower power, but a higher torque. The rotational speed and the maximum torque supplied to each gear depends on the pulse actuator described in the next section. At the end of this chapter it will be proven that a diametral pitch of 12 for each of the 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 depend on the frequency and the amplitude of the pulse. They also show that the rollover coefficient can be decreased by subtracting the pulse and that a desired trajectory can be obtained by adding the pulse to the steering wheel input. Taking this into account the pulse actuator must be able to modulate both the frequency and the amplitude of the pulse and it must be able to switch between adding and subtracting of the pulse from the steering wheel input. The mechanism 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 there is an existing mechanism able to satisfy the above constrains. Some of these mechanisms are described below. Based on the found information a mechanism is chosen. This mechanism is further designed (see section 4.4) to be able to apply a pulse with a frequency as high as possible. Mechanisms A mechanism able to modulate the frequency relatively easily by changing the rotational speed of the motor is the three-bar mechanism (see Figure 5.2 (a)) and the four-bar mechanism (see Figure 5.2 (b)). In the three-bar mechanism link 1 rotates around fixed axis A, causing link 2 to oscillate around fixed axis B. In the four-bar mechanism link 1 rotates around fixed axis A, 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 oscillation can not be changed and therefore, the amplitude of the movement can not be adjusted without 3

41 motor control. There are some mechanisms that are able to change the angle of oscillation without 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, which rotates around fixed axis A. The stroke of link 3 or the oscillation of link 2 can be changed with 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 around fixed axis A, causing link 1 to oscillate around fixed axis C. The length C-D of rocker arm 1 can 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 fixed axis C, causing link 4 to slide in guide c and causing rocker link 6 to oscillate around axis E. The angle of rotation can be varied by changing the position of point B by screw a. Note that 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, rotating around fixed axis B and is connected to slotted link 5 at point C. The rocker link is link 2 and oscillates around fixed axis B when link 1 rotates. The angle of oscillation of link 2 can be changed by changing the position of pin A by screw 3. (a) Link-gear mechanism with driven link angle of oscillation adjustment [12] (b) Lever-ratchet mechanism with variable angular velocity of the ratchet wheel [12] (c) Four-bar mechanism with a rocker arm of variable length [12] (d) Three-bar link-gear mechanism with driven link stroke adjustment [12] Figure 5.3: adjustable-amplitude mechanisms The frequency of the movements of these mechanisms can also be easily changed by changing the rotational speed of the motor. The disadvantage of these systems is however 31