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Spinning Out, With Calculus. J. Christian Gerdes Associate Professor Mechanical Engineering Department Stanford University. Future Vehicles…. Clean Multi-Combustion-Mode Engines Control of HCCI with VVA Electric Vehicle Design. Safe By-wire Vehicle Diagnostics Lanekeeping Assistance
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Spinning Out, With Calculus J. Christian Gerdes Associate Professor Mechanical Engineering Department Stanford University
Future Vehicles… Clean Multi-Combustion-Mode Engines Control of HCCI with VVA Electric Vehicle Design Safe By-wire Vehicle Diagnostics Lanekeeping Assistance Rollover Avoidance Fun Handling Customization Variable Force Feedback Control at Handling Limits
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
handwheel handwheel angle sensor handwheel feedback motor shaft angle sensor steering actuator power steering unit pinion steering rack 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
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
Lanekeeping Assistance • Energy predictions work! • Comfortable, guaranteed lanekeeping • Another example with more drama…
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
When Do Cars Spin Out? • Can we figure out when the car will spin and avoid it?
Tires • Let’s use your knowledge of Calculus to make a model of the tire…
An Observation… • A tire without lateral force moves in a straight line Tire without lateral force
An Observation… • A tire without lateral force moves in a straight line Tire without lateral force
An Observation… • A tire without lateral force moves in a straight line Tire without lateral force
An Observation… • A tire subjected to lateral force moves diagonally Tire with lateral force
An Observation… • A tire subjected to lateral force moves diagonally Tire with lateral force
An Observation… • A tire subjected to lateral force moves diagonally Tire with lateral force
An Observation… • A tire subjected to lateral force moves diagonally How is this possible? Shouldn’t the tire be stuck to the road?
Tire Force Generation • The contact patch does stick to the ground • This means the tire deforms (triangularly)
Tire Force Generation • Force distribution is triangular • More force at rear • Force proportional to slip angle initially • Cornering stiffness • Force is in opposite direction as velocity • Side forces dissipative a
Saturation at Limits • Eventually tire force saturates • Friction limited • Rear part of contact patch saturates first a Fy a
Simple Lateral Force Model • Deflection initially triangular • Defined by slip angle • Force follows deflection • Assume constant foundation stiffness cpy • qy(x) is force per unit length x = a x = -a a v(x) = (a-x) tana a qy(x) = cpy(a-x) tana
Simple Lateral Force Model • Calculate lateral force x = a x = -a a v(x) = (a-x) tana a qy(x) = cpy(a-x) tana Cornering stiffness
Tire Forces with Saturation • Tire force limited by friction • Assume parabolic normal force distribution in contact patch qz(x)
Tire Forces with Saturation • Tire force limited by friction • Assume parabolic normal force distribution in contact patch • Rubber has two friction coefficients: adhesion and sliding • Lateral force and deflection are friction limited • qy(x) <mqz(x) msqz(x) mpqz(x)
Tire Forces with Saturation • Tire force limited by friction • Assume parabolic normal force distribution in contact patch • Rubber has two friction coefficients: adhesion and sliding • Lateral force and deflection are friction limited • qy(x) <mqz(x) • Result: the rear part of the contact patch is always sliding large slip small slip msqz(x) mpqz(x)
Calculate Lateral Force xsl msqz(x) mpqz(x)
Lateral Force Model • The entire contact patch is sliding when a = asl • The lateral force model is therefore: • Figures show shape of this relationship
Lateral Force Behavior • ms=1.0 and mp=1.0 • Fiala model
Coefficients of Friction • Sliding (dynamic friction): ms = 0.8 • Many force-slip plots haveapproximately this much friction after the peak, when the tire is sliding • Seen in previous literature • Adhesion (peak friction): mp = 1.6 • Tire/road friction, tested in stationary conditions, has been demonstrated to be approximately this much • Seen in previous literature • Model predicts that these values give Fpeak / Fz = 1.0 • Agrees with expectation
Lateral Force with Peak and Slide Friction • ms=0.8 and mp=1.6 • Peak in curve • Can we predict friction on road?
How Early Can We Estimate Friction? loss of control linear nonlinear
Friction estimated about halfway to the peak – very early! Ramp: Friction Estimates linear nonlinear loss of control
Bicycle Model • Outline model • How does the vehicle move when I turn the steering wheel? • Use the simplest model possible • Same ideas in video games and car design just with more complexity • Assumptions • Constant forward speed • Two motions to figure out – turning and lateral movement
a b b ar d V af r Bicycle Model • Basic variables • Speed V (constant) • Yaw rate r – angular velocity of the car • Sideslip angle b – Angle between velocity and heading • Steering angle d – our input • Model • Get slip angles, then tire forces, then derivatives
Calculate Slip Angles a b b ar d V af r ar d+ af
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, d and b • Calculate front and rear forces from slip angles • Calculate changes in r and b
Making Sense of Yaw Rate and Sideslip • What is happening with this car?
For Normal Driving, Things Simplify • Slip angles generate lateral forces • Simple, linear tire model (no spin-outs possible) a Fy
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