John T. Cameron Pennsylvania State University Dr. Sean Brennan Pennsylvania State University

1. A Comparative, Experimental Study of Model Suitability to Describe Vehicle Rollover Dynamics for Control Design John T. Cameron Pennsylvania State University Dr. Sean Brennan Pennsylvania State University Vehicle Dynamic Modeling for the Prediction and Prevention of Vehicle Rollover

2. Outline • Goals • Analytical Vehicle Models • Experimental Model Validation • Conclusions Vehicle Dynamic Modeling for the Prediction and Prevention of Vehicle Rollover

3. Goals • Examine various vehicle models to determine the effect that different assumptions have on: • Model order • Model complexity • Number and type of parameters required • Experimentally validate the models to: • Determine model accuracy • Relate modeling accuracy to assumptions made • Determine the simplest model that accurately represents a vehicles planar and roll dynamics Vehicle Dynamic Modeling for the Prediction and Prevention of Vehicle Rollover

4. Analytical Vehicle Models • Standard SAE sign convention Vehicle Dynamic Modeling for the Prediction and Prevention of Vehicle Rollover

5. Analytical Vehicle Models • Basic Assumptions Common to All Models • All models are linear • Result: • Small angles are assumed making cos(θ)≈1, sin(θ)≈0 • Constant longitudinal velocity (along the x-axis) • The lateral force acting on a tire is directly proportional to slip angle • Longitudinal forces ignored • Tire forces symmetric right-to-left Vehicle Dynamic Modeling for the Prediction and Prevention of Vehicle Rollover

6. Analytical Vehicle Models • Model 1 – 2DOF Bicycle Model Vehicle Dynamic Modeling for the Prediction and Prevention of Vehicle Rollover

7. Analytical Vehicle Models • Model 2 – 3DOF Roll Model • Assumes the existence of a sprung mass • No x-z planar symmetry • Originally presented by Mammar et. al., National Institute of Research on the Transportations and their Security (INRETS), Versailles, France in 1999 Vehicle Dynamic Modeling for the Prediction and Prevention of Vehicle Rollover

8. Analytical Vehicle Models • Model 3 – 3DOF Roll Model • Assumes the existence of a sprung mass • x-z planar symmetry • Roll-steer influence • Originally presented by Kim and Park, Samchok University, South Korea, 2003 Vehicle Dynamic Modeling for the Prediction and Prevention of Vehicle Rollover

9. Analytical Vehicle Models • Model 3 (continued) • As a result of the assumption of roll steer, the external forces acting on the vehicle change accordingly Vehicle Dynamic Modeling for the Prediction and Prevention of Vehicle Rollover

10. Analytical Vehicle Models • Model 4 – 3DOF Roll Model • Assumes a sprung mass suspended upon a massless frame • x-z planar symmetry • No roll steer influence • Originally presented by Carlson and Gerdes, Stanford University, 2003 Vehicle Dynamic Modeling for the Prediction and Prevention of Vehicle Rollover

11. Analytical Vehicle Models • Effect of assuming force equivalence • Slightly changes plant description (i.e. eigenvalues) • Additionally, causes a higher gain in roll response from the massless frame assumption Vehicle Dynamic Modeling for the Prediction and Prevention of Vehicle Rollover

12. Model Fitting Procedures • Experimentally determine the understeer gradient to find the relationship between front and rear cornering stiffness values. Considering both frequency and time domains*: • Determine estimates on cornering stiffness values by fitting of the 2DOF Bicycle Model (Model 1). • Determine estimates on roll stiffness and damping by fitting of Models 2 – 4. * - Time domain maneuvers were a lane change and a pseudo-step Vehicle Dynamic Modeling for the Prediction and Prevention of Vehicle Rollover

13. Time Domain Fit Results Vehicle Dynamic Modeling for the Prediction and Prevention of Vehicle Rollover

14. Model Fitting Results • Results for Steering Input to Lateral Acceleration • Freq. Domain Fit Vehicle Dynamic Modeling for the Prediction and Prevention of Vehicle Rollover

15. Model Fitting Results • Results for Steering Input to Yaw Rate • Freq. Domain Fit Vehicle Dynamic Modeling for the Prediction and Prevention of Vehicle Rollover

16. Model Fitting Results • Results for Steering Input to Roll Rate • Freq. Domain Fit Vehicle Dynamic Modeling for the Prediction and Prevention of Vehicle Rollover

17. Model Fitting Results • Inconsistency in roll rate measured response does not appear at lower speeds • Better sensors are required to clarify inconsistencies in data – especially lateral acceleration and roll rate Vehicle Dynamic Modeling for the Prediction and Prevention of Vehicle Rollover

18. Remarks on Model Validation • As a result of overall accuracy and simplicity, Model 3 was chosen for further investigation. This entails: • The development of model-based predictive algorithms for rollover propensity • The development of control algorithms for rollover mitigation Vehicle Dynamic Modeling for the Prediction and Prevention of Vehicle Rollover

19. Conclusions • A relatively simple dynamic model is capable of modeling both the planar and roll dynamics of a vehicle well under constant speed conditions. • Relatively accurate measurements may be taken with inexpensive sensors • The dynamics are seen even with commercial grade sensors • Important for industry because such sensors are typically found in production vehicles • Extra care should be taken when model fitting in the time domain Vehicle Dynamic Modeling for the Prediction and Prevention of Vehicle Rollover

20. Time Response Tests • Pseudo-Step Response, 8.9 m/s, 0.09 rad amplitude, FR Params Vehicle Dynamic Modeling for the Prediction and Prevention of Vehicle Rollover

21. Time Response Tests • Pseudo-Step Response, 8.9 m/s, 0.09 rad amplitude, TR Params Vehicle Dynamic Modeling for the Prediction and Prevention of Vehicle Rollover

22. Time Response Tests • Lane Change Maneuver, 17.8 m/s, Right-to-Left, then Left-to-Right, FR Vehicle Dynamic Modeling for the Prediction and Prevention of Vehicle Rollover

23. Time Response Tests • Lane Change Maneuver, 17.8 m/s, Right-to-Left, then Left-to-Right, Time Vehicle Dynamic Modeling for the Prediction and Prevention of Vehicle Rollover

24. Experiments Performed • Determination of Understeer Gradient • Understeer gradient is a constant indicating the additional amount of steering necessary to maintain a steady-state turn per g of lateral acceleration (e.g. units are rad/g) • Provides a relationship between the front and rear cornering stiffness‘ • Lateral acceleration was measured on a 30.5 m radius circle at 6.7, 8.9, and 11.2 m/s Vehicle Dynamic Modeling for the Prediction and Prevention of Vehicle Rollover

25. Model Fitting Procedure • Step 1 – Determine understeer gradient • Plotting additional steering angle vs. lateral acceleration, the understeer gradient is simply the slope of the line Vehicle Dynamic Modeling for the Prediction and Prevention of Vehicle Rollover

26. Analytical Vehicle Models Vehicle Dynamic Modeling for the Prediction and Prevention of Vehicle Rollover