Simulation and analysis of vertical displacement characteristics of three wheels reverse trike vehicle with PID controller application

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1 Simulation and analysis of vertical displacement characteristics of three wheels reverse trike vehicle with PID controller application Wibowo, Lullus Lambang, Erick Chandra N., Nurul Muhayat, and Jaka S. B. Citation: AIP Conference Proceedings 1867, (2017); View online: View Table of Contents: Published by the American Institute of Physics Articles you may be interested in Investigation of flood routing by a dynamic wave model in trapezoidal channels AIP Conference Proceedings 1867, (2017); / Constrained H control for low bandwidth active suspensions AIP Conference Proceedings 1867, (2017); / Particle swarm optimization Genetic algorithm (PSOGA) on linear transportation problem AIP Conference Proceedings 1867, (2017); / Preface: 2nd International Conference on Mathematics Pure, Applied and Computation AIP Conference Proceedings 1867, (2017); / Aerodynamic performance of a small vertical axis wind turbine using an overset grid method AIP Conference Proceedings 1867, (2017); / Categorizing document by fuzzy C-Means and K-nearest neighbors approach AIP Conference Proceedings 1867, (2017); /

2 Simulation and Analysis of Vertical Displacement Characteristics of Three Wheels Reverse Trike Vehicle with PID Controller Application Wibowo 1,a), Lullus Lambang 1,b), Erick Chandra N. 1,c), Nurul Muhayat, 1,d), Jaka S.B. 1,e) 1) Mechanical Engineering Department, Sebelas Maret University, Surakarta 57128, Indonesia a) b) c) d) e) Abstract.The purpose of this research is to obtain a mathematical model (Full Vehicle Model) and compare the performance of passive and active suspension systems of a Three-Wheels Reverse Trike vehicle. Vehicle suspension system should able to provide good steering handling and passenger comfort. Vehicle suspension system generally only uses passive suspension components with fix spring and damper coefficients. An active suspension developed from the traditional (passive) suspension design can directly control the actuator force in the suspension system. In this paper, modeling and simulation of passive and active suspension system for a Full Vehicle Model is performed using Simulink- MATLAB software. Ziegler & Nichols tuning method is used to obtain controller parameters of Proportional Integral Derivative (PID) controller. Comparison between passive and active suspension with PID controller is conducted for disturbances input of single bump road surface profile 0.1 meters. The results are the displacement and acceleration of the vehicle body in the vertical direction of active suspension system with PID control is better in providing handling capabilities and comfort for the driver than of passive suspension system. The acceleration of 1,8G with the down time of 2.5 seconds is smaller than the acceleration of 2.5G with down time of 5.5 seconds. INTRODUCTION Automotive development leds to demands of security and comfort handling level increases. Vehicle travels on a various road profile that can affect the comfort and capability of the vehicle. The suspension system must be able to support vehicles to maneuver, carrying passengers for every different road surface conditions. Driving comfort requires soft suspension but if for transporting cargo, hard suspension may required. Good handling requires suspension settings between this two criteria [1]. Then the suspension should be adjusted for different vehicles, for example, between the race car and limousine require different settings. The main function of the suspension is to minimize the vertical acceleration transmitted to passenger. The suspension system consists of a spring, shock absorber and linkage connecting the vehicle body with the wheels. There are three types of suspension system.i.e. passive, semi-active and active suspension systems [2]. The suspension consists only springs and dampers is passive suspension whereas the suspension system with external controls are semi-active and active suspension [1]. Active suspension system can improve the ability of handling and confortably level simultaneously. An active suspension system can also produce different adjustment according to road conditions without excess the travel limit [3] This study simulates a research vehicle: UNS three-wheel reverse trike with passive suspension and active suspension with PID controller. Simulation outputs are displacement and the acceleration of the vehicle body vertical direction. Controller parameters are determined by Ziegler & Nichols tuning method. Unit-step input is used to determine the characteristics of the system. Then disturbances in the form of road surface profile is input to the system in order to investigate the performance of suspension systems. International Conference on Mathematics: Pure, Applied and Computation AIP Conf. Proc. 1867, ; doi: / Published by AIP Publishing /$

3 EQUATION OF MOTION OF VEHICLE MODEL The vehicle model is a dynamics model with 6 degrees of freedom to describe the vibration behavior of the vehicle in the vertical direction. Model 6 DOF consists of a sprung mass single (the vehicle body) connected on three unsprung mass m u1, m u2, m u3(wheels on the front left, front right, rear wheel suspensions). Sprung mass is free to perform vertical motion, pitch, and roll while the unsprung mass is free to move in the vertical direction. Thus the system enables to simulate the load forces on three tires, the acceleration and displacement, as well as roll and pitch of the vehicle body. The suspension between the sprung mass and unsprung mass is modeled as passive damper and spring elements while the tire is modeled as a linear spring without damper. The free-body diagram of three-wheel reverse trike vehicle is shown in figures 1 and 2. Fig. 1 shows a vehicle model with passive suspension, while figure 2 shows a vehicle model with an active suspension. Table 1 shows the parameters of the model. FIGURE 1. Model 6 DOF Vehicle with Passive Suspension FIGURE 2. Model 6 DOF Vehicle with Active Suspension TABLE 1. Vehicle suspension system parameter m u body mass or sprung mass (kg) m uf andm ur front and rear wheel mass or the unsprung mass (kg) I xx andi yy inertia roll and pitch moments (kgm 2 ) Z s displacement body mass or sprung mass (m) Z sfl, Z sfr, Z sr displacement of the body or sprung at each corner (m) Z ufl, Z ufr, Z ur displacement or unsprung wheel (m) w f andw r distance between the center of gravity to the front wheel-rear wheel (m) l f andl r length between the center of gravity to the front wheels and the rear wheels (m) C sf andc sr damping coefficient of front and rear suspension (Nm / s) K sf andk sr springs stiffness coefficient of front and rear suspension (N / s) K tf andk tr stiffness coefficient of front and rear tires (N / s) F pfr force control actuators at each front-right wheel suspension, F pfl force control actuators at each front-left wheel suspension force actuator control at rear wheel suspension F pr Vehicle dynamics equations in the vertical movement are obtained using Newton's second law as follows: The moment equilibrium for roll movements of sprung mass is: I xx φ s = C sf w f (Z sfr Z ufr) + C sf w f (Z sfl Z ufl) + C sr w r (Z sr Z ur) K sf w f (Z sfr Z ufr ) + K sf w f (Z sfl Z ufl ) + K sr w r (Z sr Z ur ) + w f F pfr w f F pfl + w r F pr (1) Because the value w of the rear wheels is zero, then the equation (1) becomes: I xx φ s = C sf w r (Z sfr Z ufr) + C sf w f (Z sfl Z ufl) K sf w f (Z sfr Z ufr ) + K sf w f (Z sfl Z ufl )

4 +w f F pfr w f F pfl (2) The moment equilibrium for pitch movement of sprung mass is as follows: I yy θ s = C sf l f (Z sfr Z ufr) C sf l f (Z sfl Z ufl) + C sr l r (Z sr Z ur) K sf l f (Z sfr Z ufr ) K sf l f (Z sfl Z ufl ) + K sr l r (Z sr Z ur ) + F pfr l f + F pfl l f F pr l r (3) Vehicle dynamics equations on the sprung mass vertical movement is as follows: m s Z s = C sf (Z sfr Z ufr) C sf (Z sfl Z ufl) C sr (Z sr Z ur) K sf (Z sfr Z ufr ) K sf (Z sfl Z ufl ) K sr (Z sr Z ur ) + F pfr + F pfl + F pr (4) The force equilibrium in the three unsprung mass are the following equations: m uf Z ufr = C sf (Z sfr Z ufr) + K sf (Z sfr Z ufr ) K tf Z ufr + K tf Z rfr F pfr (5) m uf Z ufl = C sf (Z sfl Z ufl) + K sf (Z sfl Z ufl ) K tf Z ufl + K tf Z rfl F pfl (6) m ur Z ur = C sr (Z sr Z ur) + K sr (Z sr Z ur ) K tr Z ur + K tr Z rr F pr (7) The position of the sprung mass at each angle variable of the vehicle given by the following equation: Z sfr = w f φ s + l f θ s + Z s (8) Z sfl = w f φ s + l f θ s + Z s (9) Z sr = w r φ s l r θ s + Z s (10) Because the value of w = 0 for the rear wheels, equation (10) for Zsr be: Z sr = l r θ s + Z s (11) Free sprung mass at each angle variable of the time derivative of position (equation (8) to (10)): Z sfr = w f φ s + l f θ s + Z s (12) Z sfl = w f φ s + l f θ s + Z s (13) Z sr = l r θ s + Z s (14) From the equations of motion, the state space equations for the system can be written as follows. X (t) = AX + BU (15) Y (t) = CX (16) where: A = State Matrix ; X (t) = State Vector ; B = Input Matrix ; U (t) = Input Vector; Y (t) = Output Equation; C = Output Matrix. The state vector X(t) is X = [X 1 X 2 X 3 X 4 X 5 X 6 X 7 X 8 X 9 X 10 X 11 X 12 ] T. The vector consists of 12 elements which is 3 DOF of body vehicle and 3DOF of each wheel suspension. Table 2 shows the components of the state vector X (t)

5 TABLE 2. Variable state of full vehicle model of three wheels reverse trike vehicle Variable Definition Variable Definition X 1 = φ s angle of rolling X 7 = φ s Free roll X 2 = θ s angle of pitch X 8 = θ s speed pitch X 3 = Z s vertical displacement X 9 = Z s vertical Speed X 4 = Z ufl vertical displacement of left front wheel X 10 = Z ufr vertical speed of left front wheel X 5 = Z ufr vertical displacement of Right front wheel X 11 = Z ufl vertical speed of right front wheel X 6 = Z ur vertical displacement of rear wheels X 12 = Z ur Vertical speed of rear wheel PROPORTIONAL-INTEGRAL-DERIVATIVE (PID) CONTROLLER PID controller works based on the value of error, the difference between the measured process variable and a desired setpoint. The controller works to minimize the error by adjusting the input process control [3]. Relations input ('error') and the output of the PID controller is shown by the equations: de(t) mv(t) = K P e(t) + K i e(t)dt + K d (17) where : dt mv(t) = output of the PID controller or manipulated variable K p = Proportional parameter constant T i = Integral parameter constant T d = Derivative parameter constant e(t) = error (difference between the set point with the actual level) K i = K p/t i K d = K p.t d Experimentally Tuning of PID Controller PID controller tuning process aims to obtain the values of a porportional (K p), an integral value (K i), and a derivative (K d) constants so that the output response meet with design objectives. Tuning methods used is the methods developed by Ziegler&Nichols. TABLE 3. Ziegler & Nichols tuning rules Controller Type K p T i T d P 05K cr 0 PI 0.45K cr 1/1.2P cr 0 PID 0.6K cr 0.5P cr 0.125P cr (a) (b) (c) FIGURE 3. Root locus of (a) system for the transfer function Zs against Fpfr; (b) system for the transfer function Zs against Fpfl (c) system for the transfer function Zsagainst Fpr

6 TABLE 4. Tuning Ziegler & Nichols Input Control Critical Values K cr Frequency ω cr(rad/sec) Critical Period P cr=2π/ω cr (sec) F pfr F pfl F pr Table 3 shows the rules of Ziegler&Nichols method in the form of a set of K p, K i, and K d which will provide a stable system. Values in the table can be used as a start points for better tuning (fine tuning). The method used to obtain the value of K CR and P CR is the method of Ziegler&Nichols second (ZN2) [4]. K CR is a critical value and the P CR is the critical period value. Determining the value of K CR and P CRusing ZN2 method is an experimental way. If a mathematical model of a system is known then the critical value K CR and the oscillation frequency ω crcan be obtained from the point where root-locus crosses the imaginary axis jω. Due to the mathematical model of three-wheeled reverse strike vehicle have been known then the root locus method can be used to obtain the critical value, K CR. MATLAB is used to generate the root-locus diagram. Furthermore, the value of P CR can be obtained from 2π / ωcr = P cr. Fig. 3 shows the root locus diagram for each transfer function of output Z s against F pfr, F pfl, and F PR. The gain obtained in each transfer function is then determined as the critical K CR. Table 4 shows K CR and P CR values determined as the intersection point of the root locus with the imaginary axis j, i.e. the imaginary part of the intersection point.k CR and P CRvalues is then used to determine the value of the PID controller parameters according to the rules of Ziegler&Nichols. Tables 5, 6 and 7 show the value of parameters PID controller for the right front wheel suspension system, the left front wheel and rear wheel, respectively. TABLE 5. Ziegler-Nichols Tuning for PID parameters of right front wheel suspension (Fpfr) Controller type Kp Ti Td P PI PID TABLE 6. Ziegler&Nichols Tuning for PID parameters of leftfront wheel suspension (Fpfl) Controller type Kp Ti Td P PI PID TABLE 7. Ziegler&Nichols Tuning for PID parameters of rear wheel suspension (Fpr) Controller type Kp Ti Td P PI PID

7 Fine Tuning PID Uses Tuning Map [18] Fine tuning to determine PID controller parameter is done by developing a tuning map. Tuning map is an array of system responses that is organized based on varying value of controller parameters. From the map, a trend of parameter changes is then determined. Fig. 4 shows the map for PID controller tuning of the right front wheel suspension system. The maximum error can be reduced by increasing the value of K p and settling time can be reduced by increasing the value of K i, except for large value of K p (1.5 K p). Furthermore, Figure 5 clearly shows the effect of changes in the value of K d which reduces settling time. The value of these parameters can be used as a PID controller parameter values. However, further tuning is still possible to achieve design objectives that is settling time of less than 2 seconds and less than 10% overshoot. FIGURE 4. Tuning map of the right front wheel suspension system

8 FIGURE 5. Effect of changes in value Kd against disturbance rejection response. Fine Tuning PID Uses Trial and Error Method Method of fine tuning PID controller parameter with trial and error method has been developed in this study. The objective of fine tuning process is to reduce settling time and rise time by maintaining steady state error at zero and overshoot value below 10%. The proses is a trial and error method based on the results of Ziegler & Nichols second method (ZN2). Fine tuning is performed according to the tendency of the effect of changing parameters as shown in the tuning map. Fine tuning procedure is as follows: 1. Adding value of K d to eliminate steady-state errors 2. Adding value of K i to reduce overshoot 3. After K i that results certain overshoot is obtained, then K d value is regulated to obtain a certain rise time. 4. Then add the value of K pto increase the rise time. The fine tuning process is stopped after a steady-state error, settling time and rise time in accordance with the desired value in the design is achieved. SIMULATION METHOD Vehicle research : Three-Wheels Reverse Trike Fig. 6 shows a research vehicle: three-wheels reverse trike used as the object of this study. Table 8 shows the list of dynamics properties of the vehicle. In this research, they are obtained by direct measurement and assumptions. FIGURE 6. Vehicle Research : three- wheels Reverse Trike

9 TABLE 8. Properties of Three-Wheels Reverse Trike Vehicle for Simulation Symbol Description values ms sprung mass 170 kg I yy The moment inertia of pitch kgm 2 I xx The moment inertia of roll kgm 2 m ufl Unsprung mass of left front wheel 45 kg m ur Unsprung mass of rear wheel 80 kg m ufr unsprung mass of right front wheel 45 kg K sfl spring stiffness coefficient of the left front wheel suspension N/m K srr Spring stiffness coefficient of the rear wheel suspension N/m K sfr Spring stiffness coefficient of the right front wheel suspension N/m C sfl Damping coefficient of left front wheel suspension 1200 Ns/m C srr damping coefficient of rear wheel suspension 2000 Ns/m C sfr Damping coefficient of right front wheel suspension 1200 Ns/m K tfl stiffness coefficient of left front tire N/m K tr stiffness coefficient of rear tire N/m Input profile single road bump A function of road surface profiles are used as input to the suspension system in a simulation in order to evaluate the response of the suspension system. The function of a bumpy road surface consists of a single wave with an amplitude of a [3]. In this simulation, the amplitude which represents the bump height of 0.1 meters. Function bumpy road profile is as follows: FIGURE 7. Street profile of single bump a(1 cos8πt)/2, 0.5 t 0.75 r(t) = f(x) = { 0, otherwise (18) Fig. 7 shows the road profile with a single bump according to equation (18). The figure shows that the waves began to occur between 0.5 seconds to 0.75 seconds, from the early start of the vehicle running

10 RESULTS AND DISCUSSIONS Response of Passive (Open-Loop) Suspension System or System without A Controller Open-loop system response to a unit step input is shown in Figure 8 for input on the right front wheel, left front wheel and rear wheel, respectively. Fig. 8 (a) and (b) show the same response as the suspension system used is the same. While figure 8 (c) shows a damped response since it uses the damping of 2000 Ns / m compared to the damping of 1200 Ns/m is used for the front wheel suspensions. The dynamic response of the passive damping system has a large percentage of overshoot and settling time, then to correct this response, next system of closed loop system with PID controller is developed. (a) (b) (c) FIGURE 8. Response to unit-step input of passive suspension system (open-loop) (a) front-right wheel, (b) the left front wheel, (c) the rear wheel Response System with PID Controller Parameters Determined by The ZIEGLER & NICHOLS Method Fig. 9(a), (b) and (c) shows the step response unit for each of the right front wheel suspension system, the left front wheel and rear wheel. These charts clearly shows that the controller has been able to reduce overshoot. (a) (b) (c) FIGURE 9. Response active suspension system using input with Ziegler & Nichols tuning method of (a) the front-right wheel (b) the left front wheel (c) the rear wheel. Response of Suspension System with PID Contoller Parameters Determined with Fine Tuning Fine tuning by trial and error method is performed to reduce settling time and rise time by maintaining steady state error at zero and overshoot value below 10%. In this fine tuning, error remains based on the results of tuning Ziegler

11 & Nichols as the reference to increase the value of Kp in order to increase the rise time. Adding of K d will increase the overshoot, and adding the value of Ki will eliminate steady state errors [6]. Table 9 shows the value of the PID controller parameters obtained with fine tuning using trial and error method. It can be seen that the value of the PID controller parameters in the same order both for passive and active suspension system. Fig. 10 shows the response to a unit step input for the right front wheel suspension system, the left front wheel and rear wheel, respectively. The figures show that the controller with fine tuning parameter has the characteristics of a rise time that is faster than previous controllers and overshoot is in acceptable percentage of less than 10%. Table 10 is a summary of the performance of transient dynamic systems of vehicles on the input unit step. The system includes a passive suspension system (open-loop) and active suspension system with and without advanced tuning. The performance of the system be better to use a controller with advanced tuning in terms of rise time and settling time and overshoot percentage is still acceptable at less than 10%. TABLE 9. Parameters obtained by fine Tuning By using Trial and Error method System Parameter tuning Value Ziegler&Nichols fine Tuning value bytrial and Error method Control Input F pfr K p K i e+006 K d e+004 Control Input F pfl K p K i e+006 K d e+004 Control Input F pr K p K i e+005 K d e+003 (a) (b) (c) FIGURE 10. Response active suspension system for step unit input with controller parameter determined using fine tuning (green lines) and Ziegler & Nichols tuning (blue line): (a) the front-right wheel (b) the left front wheel (c) the rear wheels

12 TABLE 10. Characteristics response system for unit step-input. Response characteristics Open-loop Tuning Ziegler&Nichols Fine Tuning Front right wheel: Rise time Settling time Overshoot Peak 1.52E Peaktime Front left wheel: Rise time Settling time Overshoot Peak 1.52E Peaktime Rear wheel: Rise time Settling time Overshoot Peak 2.36E Peaktime Comparisson on the performance of the controller with Ziegler & Nichols tuning and fine tuning shown in Table 11. It can be seen a trend chart after fine tuning that is closer to input step function value. This is because the rise time decreases and the value of settling time is reduced to less than two second for all wheels and although the percentage of overshoot increase but the value still below 10%. ANALYSIS OF SYSTEM RESPONSE TO INPUT DISTURBANCES ROAD PROFILE Dynamics simulation was conducted to determine the system responds to a disturbance of profile of a road bump as shown in equation (18). Input of road profile function is applied to one of the wheels and all-wheel of the threewheeled vehicles. The suspension system used is a suspension system with controller defined with fine tuning. The simulation results are the following

13 TABLE 11. System Response for unit step input Response characteristics Change in (%) compared to Open-loop system Ziegler and Nichols Tuning fine Tuning Front-right wheel: Rise time increase 2.40 increase Settling time 5.57 increase decrease Overshoot decrease decrease Peak increase increase Peaktime increase increase Front-left wheel: Rise time increase 2.40 increase settling time 5.57 increase decrease Overshoot decrease decrease Peak increase increase Peaktime increase increase Rear wheel: Rise time increase increase settling time increase decrease Overshoot decrease decrease Peak increase increase Peaktime increase increase Vertical Displacement of Sprung Mass Subjected to Disturbance (road profile) on One Wheel. Fig. 11 shows the response of the suspension system of each wheel to road profile function (disturbance). This figure shows the profile functions as a green line and the response of the vertical displacement of the sprung mass / vehicle body as light blue lines (passive suspension) and the red line (active suspension). The figure clearly shows the response of the passive damping system of lower amplitude maximum deviation and thus the system is more damped. (a) (b) (c) FIGURE 11. Vertical displacement of the sprung mass of passive suspension (red lines) and active suspension (green lines) when the suspension system is subjected to road profile (disturbance). Vertical Displacement of The Sprung Mass When Subjected to Disturbance (road profile) on All Wheels Simultaneously Figure 12 shows the response of the vehicle's suspension system to the input of road profile function (disturbance). This figure clearly shows the response of the active suspension system with lower amplitude maximum deviation than passive suspension system and thus the system is more damped. From Figures 16 and 17, it can be seen that active suspension is able to improve system performance in reducing the displacement of the sprung mass vertical direction

14 Furthermore, Table 12 shows a large reduction of the maximum vibration amplitude value when using the active suspension system. FIGURE 12. Vertical displacement of the vehicle body (sprung, Zs) : the passive suspension (red lines) and active suspension (green lines) when all the wheels pass 10 cm high of road profile (blue lines) at the same time TABLE 12. Reduction of vertical displacement of sprung mass Applied disturbance Peak displacement of the sprung mass (meters) pasive active % reduction Front right wheel Front left wheel Rear wheel All wheels ANALYSIS OF HUMAN COMFORTABLY ON A VERTICAL ACCELERATION Figure 13 shows a graph of Resistance of human body againts an acceptable Linear Acceleration [7]. This graph can be interpreted as the level of human comfort when driving. In this study the acceleration of the vertical direction (up and down) are used in the analysis of the performance of passive and active suspension systems. Acceleration with a large amplitude within a certain limit can be acceptable if it occurs in a short duration

15 FIGURE 13. Human body comfortable against an acceptable linear acceleration [7] Figure 14 shows the acceleration of the vehicle body vertical direction using passive suspension (orange lines) and active suspension (light blue lines). Simulation for input disturbance on the right front wheel. The value of maximum acceleration amplitude and down time are shown in Table 12 for a suspension system of passive and active suspension system. FIGURE 14. Vertical directions acceleration of vehicle body using passive suspension (orange lines) and active suspension (light blue lines) CONCLUSIONS Open-loop suspension systems have a steady-state error values of 1, 5.1 and 1.8 x 10-5 and the settling time is longer than 2 seconds. It means that more than 2 seconds is taken by the vehicle body back to a stable. Percentage overshoot of the vertical body displacement is still above 10%. Response to input step of active suspension system of all wheels show small steady-state error, rise time reduces to 1.796,1.7961, and sec for the front-right wheel, front-left wheel, and the rear wheel, respectively. It means the system achieves the design objective i.e. rise time less than two secs. Percentage overshoot reduces by 80.4%, 80.4%, and 89% from open-loop system for the front-right wheel, front-left wheel and rear wheel suspension. The simulation with road profile input shows that active suspension system is able to reduce significantly vibration of the vehicle body. This can be seen that the vertical displacement of vehicle body were reduced by 80% when disturbance is applied on one front wheel (right or left), and 49% when applied on the rear wheels, and 93% when applied simultaneously on all wheels. Active suspension is able to adjust in real-time on the condition of the road profile. Vertical acceleration of vehicle body is analyzed by using the graph of capability of human body for receiving acceleration. It was found that the active suspension system is able to significantly reduces vertical acceleration of 1.8G with a time of 2.5 seconds system compared with of 2.5G with a time of 5.5 seconds of passive suspension system

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