Vehicle Dynamics – It’s all about the Calculus… J. Christian Gerdes Associate Professor Mechanical Engineering Department Stanford University

Dynamic Design LabStanford University- 2 Future Vehicles… Safe By-wire Vehicle Diagnostics Lanekeeping Assistance Rollover Avoidance Fun Handling Customization Variable Force Feedback Control at Handling Limits Clean Multi-Combustion-Mode Engines Control of HCCI with VVA Electric Vehicle Design

Dynamic Design LabStanford University- 3 Electric Vehicle Design How do we calculate the 0-60 time?

Dynamic Design LabStanford University- 4 Basic Dynamics Newton’s Second Law With Calculus If we know forces, we can figure out velocity

Dynamic Design LabStanford University- 5 What are the Forces? Forces from: Engine Aerodynamic Drag Tire Rolling Resistance

Dynamic Design LabStanford University- 6 Working in the Motor Characteristics

Dynamic Design LabStanford University- 7 Working in the Motor Characteristics

Dynamic Design LabStanford University- 8 Some numbers for the Tesla Roadster From Tesla’s web site: m = mass = 1238 kg R gear = final drive gear ratio = 8.28 A = Frontal area = Height*width Overall height is 1.13m Overall width is 1.85m This gives A = 2.1m 2 but the car is not a box. Taking into account the overall shape, I think A = 1.8 m 2 is a better value to use. C D = drag coefficient = This comes from the message board but seems reasonable

Dynamic Design LabStanford University- 9 More numbers for the roadster From other sources r wheel = wheel radius = 0.33m (a reasonable value) Frr = rolling resistance = 0.01*m*g For reference, see:  = air density = 1.2 kg/m 3 Density of dry air at 20 degrees C and 1 atm To keep in mind: Engine speed w is in radians/sec The Tesla data is in RPM 1 rad/s =.1047 RPM (or 0.1 for back of the envelope calculations) 1mph = m/s

Dynamic Design LabStanford University- 10 Motor issues The website lists a motor peak torque of 375 Nm up to 4500RPM. This doesn’t match the graph. They made changes to the motor when they chose to go with a single speed transmission. I think the specs are from the new motor and the graph from the old one. Here is something that works well with the new specs:

Dynamic Design LabStanford University- 11 Results of my simulation Pretty cool – it gives a 0-60 time of about 3.8s Tesla says “under 4 seconds” Top speed is 128 mph (they electronically limit to 125)

Dynamic Design LabStanford University- 12 P1 Steer-by-wire Vehicle “P1” Steer-by-wire vehicle Independent front steering Independent rear drive Manual brakes Entirely built by students 5 students, 15 months from start to first driving tests steering motors handwheel

Dynamic Design LabStanford University- 13 Future Systems Change your handling… … in software Customize real cars like those in a video game Use GPS/vision to assist the driver with lanekeeping Nudge the vehicle back to the lane center

Dynamic Design LabStanford University- 14 Steer-by-Wire Systems Like fly-by-wire aircraft Motor for road wheels Motor for steering wheel Electronic link Like throttle and brakes What about safety? Diagnosis Look at aircraft handwheel handwheel angle sensor handwheel feedback motor steering actuator shaft angle sensor power steering unit pinion steering rack

Dynamic Design LabStanford University- 15 Bicycle Model Basic variables Speed V (constant) Yaw rate r – angular velocity of the car Sideslip angle  – Angle between velocity and heading Steering angle  – our input Model Get slip angles, then tire forces, then derivatives ff rr   V ba r

Dynamic Design LabStanford University- 16 Vehicle Model Get forces from slip angles (we already did this) Vehicle Dynamics This is a pair of first order differential equations Calculate slip angles from V, r,  and  Calculate front and rear forces from slip angles Calculate changes in r and 

Dynamic Design LabStanford University- 17 Calculate Slip Angles ff rr   V ba r  f rr

Dynamic Design LabStanford University- 18 Lateral Force Behavior  s =1.0 and  p =1.0 Fiala model

Dynamic Design LabStanford University- 19 When Do Cars Spin Out? Can we figure out when the car will spin and avoid it?

Dynamic Design LabStanford University- 20 linearnonlinear Comparing our Model to Reality loss of control

Dynamic Design LabStanford University- 21 Lanekeeping with Potential Fields Interpret lane boundaries as a potential field Gradient (slope) of potential defines an additional force Add this force to existing dynamics to assist Additional steer angle/braking System redefines dynamics of driving but driver controls

Dynamic Design LabStanford University- 22 Lanekeeping on the Corvette

Dynamic Design LabStanford University- 23 Lanekeeping Assistance Energy predictions work! Comfortable, guaranteed lanekeeping Another example with more drama…

Dynamic Design LabStanford University- 24 Handling Limits What happens when tire forces saturate? Front tire Reduces “spring” force Loss of control input Rear tire Vehicle will tend to spin Loss of stability handling limits linear region Is the lanekeeping system safe at the limits?

Dynamic Design LabStanford University- 25 Countersteering Simple lanekeeping algorithm will countersteer Lookahead includes heading error Large heading error will change direction of steering Lanekeeping system also turns out of a skid Lateral error Projected error Example: Loss of rear tire traction

Dynamic Design LabStanford University- 26 Lanekeeping at Handling Limits

Dynamic Design LabStanford University- 27 Video from Dropped Throttle Tests

Dynamic Design LabStanford University- 28 Controller countersteers to prevent spinout Lanekeeping ActiveLanekeeping Deactivated Yaw Stability from Lanekeeping

Dynamic Design LabStanford University- 29 Controller response to heading error prevents the vehicle from spinning A Closer Look

Dynamic Design LabStanford University- 30 Conclusions Engineers really can change the world In our case, change how cars work Many of these changes start with Calculus Modeling a tire Figuring out how things move Also electric vehicle dynamics, combustion… Working with hardware is also very important This is also fun, particularly when your models work! The best engineers combine Calculus and hardware