Phase Plane Analysis for Vehicle Handling and Stability

Transcription

1 International Journal o Computational Intelligence Sstems, Vol. 4, No. 6 (December, 2), Phase Plane Analsis or Vehicle Handling and Stabilit ZHANG Hao, LI Xian-sheng, SHI Shu-ming *, LIU Hong-ei, GUAN Rachel 2,LIU Li (. College o Traic, Jilin Universit, No Renmin Street, Changchun, 322, China 2. Minnesota Department o Transportation, 395 John Ireland Blvd. St. Paul, 5555, America) Abstract Nonlinear stabilit analsis o phase plane is one o the most classic and eective methods. Based on established two degrees o reedom (2 DOF) vehicle model, combined with magic ormula tire mode, phase plane analsis is used to stud vehicle motion characteristics in various initial conditions. In addition, equilibrium point and phase plane trajectories changes are analzed under the condition o traveling in a straight line, circular motion and sine wave input. The results denote that vehicle stead state can be well determined according to analsis changes o phase plane trajector and state variables. Namel, phase plane analsis can be well used to analze and evaluate the trend o variables and vehicle motion characteristics. Kewords: Nonlinear Analsis, Handling and Stabilit, Phase Plane Analsis, Magic Formula. Introduction Phase plane analsis is one o the most classic and eective methods in nonlinear stabilit analsis. This method reerences the idea o nonlinear dnamics. Man dnamic characteristics o sstem can be obtained through the observation and analsis on the phase trajector o phase plane -5, or example, the changes o sstem stabilit region can be represented b the changing trend o phase trajectories, the change o sstem stabilit during stead motion can be caused b the change o equilibrium point on phase plane. Phase plane analsis method can show sstem dnamic perormance characteristics, it is widel used in vehicle handling and stabilit analsis 5-8. Dierent phase plane analsis methods are established b researchers, such as: the aw plane established b Sacks 9, ront and rear side slip angle phase plane established b Pacejka, ratio o turning kinetic energ and orward driving kinetic energ phase plane used to evaluate steering perormance established b Konghui Guo. Konghui Guo ocused on energ phase plane transient response o a variet o tests and results o quasi-stead explanation, evaluation and comparison. Two parameters o phase plane were the dimensionless center o aw rate and side slip angle. Inagaki 3,4 proposed a phase plane which constitutes a state variable with vehicle bod side slip angle and its changing rate( β β ) and vehicle bod side slip angle and aw rate ( β r ),and studied the dnamic characteristics o 2 DOF vehicle sstem during its steering b using phase plane analsis method based on nonlinear tire model (magic ormula).young Eun Ko and Jang Moo Lee 4 identiied stabilit region o phase plane trajector in β r phase plane with dierent longitudinal speed and ront wheel angle input based on 2 DOF vehicle model and topological theor. A. Stotsk and X. Hu established the equation o 2 DOF steering dnamic sstem to determine the sstem's Lapunov unction, and obtained the analtical expression o state variables (aw rate and steering wheel angle) in border o stabilit region b using the expression o the cubic term tire orce. John Samsundar 4 obtained the analtical expression o the stable region o the steering dnamic sstem b using the Lapunov second method and the expression o cubic term tire orce, and obtained * Corresponding author: shishuming@jlu.edu.cn

2 Hao Zhang et al. conservative and elliptical stabilit boundar o the sstem b numerical simulation. Shuming Shi and Zhenong Mao 2-3 obtained the tire lateral orce using linear equations itted b irst order Talor expansion ormula express nonlinear expression. The scope o tire orce is extended to the whole nonlinear region, which can meet vehicle in high speed and emergenc steering. Other research content in the phase plane analsis o vehicle handling and stabilit is equilibrium point. Nonlinear sstem 4 generall has more than one equilibrium point. The results o Inagaki 3,4 show that when vehicle dnamic sstem parameters (longitude speed and ront wheel angle) change, sstem equilibrium point moves along with corresponding changes. However, when ront wheel angle increases to a certain extent, a stable ocus point and an unstable saddle point disappear, another unstable saddle point is let. Inagaki 3 and Ono 5,6 obtain the same conclusion, vehicle sstem has three equilibrium points, one point is stable ocus point, and the other two are unstable saddle points. Catino 7 uses MATCONT sotware also obtains similar conclusion. Young 5 and Nguen 8 also obtained a similar biurcation diagram, but their analsis is local eective. Shuiwen Shen 9 extended simpliied magic ormula tire mode to the whole nonlinear region, obtains three singular points, a stable ocus and two unstable saddle points using geometric method. Sstem stable motion characteristics can be obtained b analzing the change o stable equilibrium point. The border o sstem stable region can be obtained b analzing the distribution o unstable equilibrium point. Thereore, in order to enhance the understanding o vehicle planar motion stabilit, improve traic saet 2,2, based on 2 DOF vehicle model, phase plane analsis is used to stud vehicle motion characteristics under the condition o traveling in straight line, circular motion and sine wave input motion b using magic ormula tire model in this paper. 2. Establishment o vehicle nonlinear model 2.. Vehicle model establishment 2 DOF vehicle model is classical sstem equation used in handling and stabilit commonl. In this sstem equation, longitudinal vehicle speed is assumed as a constant, and tire longitudinal orce and air resistance are ignored,as shown in Fig.. m( v + v xr) = 2Fs cosδ + 2Fsr () I zr = 2Fs cosδ l 2Fsrlr Where, m vehicle mass 64kg; v vehicle lateral velocit m/s; v x vehicle longitudinal velocit m/s; r vehicle aw velocit rad/s; ront wheel steering angle rad; F s ront wheel lateral orce N, F sr ront wheel lateral orce N, I z vehicle aw moment o inertia 29kg m 2 ; l distance between ront axle and centroid.m; l r distance between rear axle and centroid.4m. Y O F sr F sr v 2 v l r r F s β v vx ψ l Fig.. Vehicle Model Side slip angles o both ront and rear tires are given as ollows: v + r l α = δ arctan vx (2) v r lr α = r arctan vx 2 DOF vehicle sstem dnamics equations can be simpliied as ollows: Fs cosδ + Fsr v = vxr m (3) Fs l cosδ Fsrlr r = I 2.2. Tire model Lateral tire orce has important impact on the vehicle nonlinear dnamics sstem stabilit analsis. The common tire orce models mainl include Konghui Guo s Unitre model 22, Pacejka s magic ormula (Magic Formula) 23,24, tire model established b Gim z v F s x X 8

3 Phase Plane Analsis or Vehicle G 25,26, and tire model established b Fiala E 27, where the magic ormula model is ormulated: = Dsin C arctan Bα E Bα arctan Bα (4) F s ( ( ( ))) Where B, C, D, E is coeicients, F is the lateral tire orce, and α is side slip angle. Lateral Tire Force F/N Front tire Rear tire Side Slip Angle α/rad Fig. 2. Lateral tire orce are local view o it. In those igures v and r taking their initial value rom the interval o (,) m/s and (,) rad/s and the initial longitudinal velocit being 25m/s and 35m/s, respectivel. Simulation time is 2s. There are three equilibrium points in phase plane, one is stable equilibrium point, and the others are unstable equilibrium point There is a strip stable region in phase plane, and point in this region could return to stable equilibrium point ater a certain time, but the outside can t return to the stable region 29,33. These results are used to distinguish the stable region and unstable region o 2 DOF vehicle model. It can be seen rom the phase plane, the middle part o vehicle is stabilit region. In this region, variables can return to the inal point o the inal stead state that is the value o lateral velocit and aw acceleration is zero. Moreover, the range o stable region is graduall shrinking while longitudinal velocit is increasing. Table Magic ormula model parameters Tire Coeicient B C D E Front tire Rear tire The coeicients are listed in Table 28, while Fig. 2 graphicall shows the relationship between the lateral tire orces and side slip angles or the respective ront and rear tires. 3. Phase plane Portraits o 2DOF In 2 DOF vehicle model, ront steering wheel angle is used as external input variable, it is an important actor in sstem equation. When ront steering wheel angle input dierent values, vehicle dnamics perormance is ver dierent, such as i the ront steering wheel angle is dierent, then trajector and equilibrium point have dierent values. Vehicle dnamics is analzed with dierent ront wheel angles (e.g. zero, a non-zero constant, sine wave input). 3.. Phase plane portraits with the variation o initial longitudinal velocit Fig. 3 and Fig. 4 show the vehicle trajectories, where Fig. (a) o Fig. 3 and Fig. 4 are global view o v -r phase plane trajectories and Fig. (b) o Fig. 3 and Fig (a) Global view o v -r phase plane trajectories (b) Local view o v -r phase plane trajectories Fig. 3. Lateral tire orce 8

4 Hao Zhang et al (a) Global view o v -r phase plane trajectories Vehicle movement will transit to stead state circular motion when ront wheel steering angle is non-zero constant value. Vehicle dnamics perormance will be analzed in dierent initial states. Fig. 5 and Fig. 6 show local phase plane trajectories on v -r plane when lateral velocit is 25m/s Simulation time is 2s. Front wheel steering angle is.rad and.5rad, respectivel. When ront wheel steering angle is small (e.g..rad), the sstem has stable equilibrium point, however when the angle increases to a certain extent(e.g..5rad), sstem's stable equilibrium point disappears and it is in unstable state o motion (b) Local view o v -r phase plane trajectories Fig. 4 v -r phase plane with initial longitudinal velocit being v x =35m/s Fig. 6. Local phase plane trajectories onto v -r plane with being.5 rad 3.2. Phase plane portraits with the variation o ront wheel steering angle 3.3. Front wheel steering angle sine wave input =.2*sin(2*π*.4*t)rad =.5*sin(2*π*.4*t)rad Front Wheel Steering Angle /rad Fig. 5. Local phase plane trajectories onto v -r plane with being. rad Fig. 7 Front wheel steering angle sine wave input 82

5 Phase Plane Analsis or Vehicle Fig. 7 represents the results o ront wheel steering angle sine wave inputs. All o requencies o the sine waves are.4hz, and the amplitudes are.2rad and.5rad, respectivel. Fig. 8 and Fig. 9 show phase plane trajectories with the amplitude o being.2rad and.5rad respectivel, and the lateral velocit is 25m/s Fig. 8 Local view o v -r phase plane trajectories with the amplitude o being.2rad Equilibrium position and the corresponding stabilit region will disappear. 4. Analsis o Results 4.. Front wheel steering angle constant inputs A trajector on the phase plane is analzed or the sake o analsis the change o phase trajector and corresponding vehicle motion. Vehicle initial state is v x =25m/s, v =r= with dierent ront wheel steering angle inputs (a) v -r phase plane trajectories Fig. 9 Local view o v -r phase plane trajectories with the amplitude o being.5rad When ront wheel steering angle is sine wave input, vehicle sstem's equilibrium position is not a ixed point, but a trajector. I vehicle sstem is stable, state variables and ront wheel angle change are consistent and show a periodic. There is an equilibrium position and stabilit region when amplitude o ront wheel steering angle is small (e.g. A=.2rad). But amplitude o ront wheel steering angle increases to a certain extent (e.g. A=.5rad), vehicle sstem will not exist =.rad =.rad (b) state variables Fig. Phase plane trajectories and state variables with ront wheel steering angle being.rad Fig. and Fig. show phase plane trajectories and state variables with ront wheel steering angle being.rad and.5rad respectivel and the lateral velocit is 25m/s. 83

6 Hao Zhang et al. The results show 2 DOF vehicle model stable and unstable phase plane and state variable time domain characteristic. When the steering sstem is stable (e.g. =.rad), the sstem can return to the stable equilibrium point, and the inal state variables v and r return to the constant stabilit values. When the steering sstem is unstable (e.g. =.5rad), state variables v and r grow rapidl and vehicle sstem loses stabilit =.2*sin(2*π*.4*t)rad Initial Point =.5rad Initial point (a) v -r phase plane trajectories / (m/s) (a) v -r phase plane trajectories Time t / s Yaw Rate r / (rad/s) 3 2 =.5rad =.5rad Time t / s (b) state variables Fig. Phase plane trajectories and state variables with ront wheel steering angle being.5rad =.2*sin(2*π*.4*t)rad =.2*sin(2*π*.4*t)rad (b) state variables Fig. 2 Phase plane trajectories and state variables with the amplitude o being.2rad =.5*sin(2*π*.4*t)rad Initial Point Front wheel steering angle sine wave input In order to analze vehicle movement characteristics o stable and unstable, a comparative analsis is studied with dierent ront wheel angle sine wave inputs when vehicle initial state is v x =25m/s, v =r= (a) v -r phase plane trajectories 84

7 Phase Plane Analsis or Vehicle =.5*sin(2*π*.4*t)rad =.5*sin(2*π*.4*t)rad (b) state variables Fig. 3 Phase plane trajectories and state variables with the amplitude o being.5rad The results show 2 DOF vehicle model stable and unstable phase plane and state variable requenc domain characteristic. When steering input is small (e.g. A=.2rad), vehicle sstem can return to stable equilibrium point, and changes o state variables v and r are stabilit. When the steering input is large (e.g. A=.5rad), state variables v and r grow rapidl, the sstem is unstable. This can be used to judge sstem stabilit. Form the results o phase plane analsis under dierent various initial conditions (in a straight line, circular motion and sine wave input), 2 DOF vehicle plane phase can analze vehicle handling and stabilit characteristics and state variables change. In other words, the proposed model and the used method can solve the vehicle handling and stabilit problem. 5. Conclusion () Local view o phase plane trajectories relects the changes o sstem stabilit region; it can be used to analze vehicle handling and stabilit characteristic under dierent various initial conditions. (2) A trajector on the phase plane can be use to analze the change o phase trajector and corresponding vehicle motion. The change o equilibrium point on phase plane relects sstem stabilit change in dierent motion. (3) When phase trajector returns to stead state, the state variables tend to a constant value, and when phase trajector can t return to stead state, the state variables increase rapidl. Thereore, according to the change o state variables can be used to judge sstem stabilit. Based on the above analsis, 2 DOF vehicle plane phase can analze vehicle handling and stabilit characteristics and state variables change. This las the oundation and theor guideline or the depth stud o related vehicle control problem. There exits some limitations with the proposed method. For example, in 2 DOF vehicle model, longitudinal movement is constrained, this model can onl partl express vehicle planar motion dnamic characteristics. So it can t be used to comprehensivel and detailed analze vehicle stabilit in high speed and emergenc steering, which will be the topics o uture works. Acknowledgements This work is supported b the National Ke Technologies R&D Program during the th Five-Year Plan Period (Project No. 29BAG3A4), Graduate Innovation Fund o Jilin Universit (Project No. 248). Reerences. Stotsk, A. and Hu, X. Stabilit analsis o robustl decoupled car steering sstem with nonlinear tire model. IEEE 36th Conerence on Decision and Control, 997, G Shoji Inagaki, Lkuo Kushiro, Masaki Yamamoto. Analsis on Vehicle Stabilit in Critical Cornering Using Phase-Plane Method. SAE paper, Yoshiuki, Kenji Tozu, Noriaki Hattori, and Masakazu Sugisawa. Improvement o Vehicle Directional Stabilit or Transient Steering Maneuvers Using Active Brake Control. SAE paper Young Eun Ko, Jang Moo Lee. Estimation o the stabilit region o a vehicle in plane motion using a topological approach. Int. J. o Vehicle Design, 22, 3(3), John Samsundar and Jere C Huston. Estimating lateral stabilit region o a nonlinear 2 degree-o-reedom vehicle. SAE Paper H. B. Pacejka, Nonlinearities in road vehicle dnamics, Vehicle Sstem Dnamics, vol. 5, pp , Vinent Nguen. Vehicle handling, stabilit, and biurcation analsis or nonlinear vehicle models. USA, Washington D. C. Universit o Marland, Shuiwen Shen, Jun Wang, Peng Shi. Nonlinear dnamics and stabilit analsis o vehicle plane motions. Vehicle Sstem Dnamics, 27, 45(): Sacks, H.K. and Digh, M. Automobile Stabilit-A Stud o the Domains o Attraction. The Dnamics o Vehicles 85

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