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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 Fun By-wire Vehicle Diagnostics Lanekeeping Assistance Rollover Avoidance Handling Customization Variable Force Feedback Control at Handling Limits Stanford University -2 Dynamic Design Lab Future Systems Change your handling… … in software Customize real cars like those in a video game Stanford University Use GPS/vision to assist the driver with lanekeeping Nudge the vehicle back to the lane center -3 Dynamic Design Lab Steer-by-Wire Systems Like fly-by-wire aircraft Motor for road wheels Motor for steering wheel Electronic link handwheel handwheel angle sensor handwheel feedback motor Like throttle and brakes shaft angle sensor What about safety? Diagnosis Look at aircraft steering actuator power steering unit pinion steering rack Stanford University -4 Dynamic Design Lab 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 Stanford University -5 Dynamic Design Lab Lanekeeping on the Corvette Stanford University -6 Dynamic Design Lab Lanekeeping Assistance Stanford University Energy predictions work! Comfortable, guaranteed lanekeeping Another example with more drama… -7 Dynamic Design Lab P1 Steer-by-wire Vehicle “P1” Steer-by-wire vehicle Independent front steering Independent rear drive Manual brakes steering motors handwheel Entirely built by students 5 students, 15 months from start to first driving tests Stanford University -8 Dynamic Design Lab When Do Cars Spin Out? Can we figure out when the car will spin and avoid it? Stanford University -9 Dynamic Design Lab Tires Let’s use your knowledge of Calculus to make a model of the tire… Stanford University - 10 Dynamic Design Lab An Observation… A tire without lateral force moves in a straight line Tire without lateral force Stanford University - 11 Dynamic Design Lab An Observation… A tire without lateral force moves in a straight line Tire without lateral force Stanford University - 12 Dynamic Design Lab An Observation… A tire without lateral force moves in a straight line Tire without lateral force Stanford University - 13 Dynamic Design Lab An Observation… A tire subjected to lateral force moves diagonally Tire with lateral force Stanford University - 14 Dynamic Design Lab An Observation… A tire subjected to lateral force moves diagonally Tire with lateral force Stanford University - 15 Dynamic Design Lab An Observation… A tire subjected to lateral force moves diagonally Tire with lateral force Stanford University - 16 Dynamic Design Lab An Observation… A tire subjected to lateral force moves diagonally How is this possible? Shouldn’t the tire be stuck to the road? Stanford University - 17 Dynamic Design Lab Tire Force Generation The contact patch does stick to the ground This means the tire deforms (triangularly) Stanford University - 18 Dynamic Design Lab Tire Force Generation a Fy Caa Stanford University 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 - 19 Dynamic Design Lab Saturation at Limits a Eventually tire force saturates Friction limited Rear part of contact patch saturates first Fy a Stanford University - 20 Dynamic Design Lab Simple Lateral Force Model x=a x = -a a Deflection initially triangular Defined by slip angle v(x) = (a-x) tana a Force follows deflection qy(x) = cpy(a-x) tana Stanford University Assume constant foundation stiffness cpy qy(x) is force per unit length - 21 Dynamic Design Lab Simple Lateral Force Model x=a x = -a a Calculate lateral force a Fy q y ( x)dx a a v(x) = (a-x) tana a qy(x) = cpy(a-x) tana Stanford University c py (a x) tan adx a 2c py a 2 tan a Ca tan a Cornering stiffness - 22 Dynamic Design Lab Tire Forces with Saturation Tire force limited by friction Assume parabolic normal force distribution in contact patch qz(x) Stanford University - 23 Dynamic Design Lab 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 msqz(x) mpqz(x) Lateral force and deflection are friction limited qy(x) <mqz(x) Stanford University - 24 Dynamic Design Lab 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 msqz(x) mpqz(x) Lateral force and deflection are friction limited qy(x) <mqz(x) Result: the rear part of the contact patch is always sliding large slip Stanford University small slip - 25 Dynamic Design Lab Calculate Lateral Force Fy q y adhesion ( x)dx q y ( x)dx sliding a xsl xsl a c py (a x) tan adx m s q z ( x)dx xsl 3Fz a 2 x 2 q z ( x) 2 4a a msqz(x) q y ( xsl ) m p q z ( xsl ) mpqz(x) Stanford University - 26 Dynamic Design Lab Lateral Force Model The entire contact patch is sliding when a asl 3m p Fz tan a sl Ca The lateral force model is therefore: 2 Ca Ca tan a Fy (a ) 3m p Fz 3 m C 2 s tan a tan a a 1 2m s tan 3 a 2 2 m 9 m Fz 3m p p p m s Fz sgn a a a sl a a sl Figures show shape of this relationship Stanford University - 27 Dynamic Design Lab Lateral Force Behavior ms=1.0 and mp=1.0 Fiala model 1 F/Fz 0.9 tp/t p0 0.8 0.6 0.5 F/F z and t /t p p0 0.7 0.4 0.3 0.2 0.1 0 0 0.5 1 1.5 C qtan a 2 2.5 3 a m p Fz Stanford University - 28 Dynamic Design Lab Coefficients of Friction Sliding (dynamic friction): ms = 0.8 Adhesion (peak friction): mp = 1.6 Many force-slip plots have approximately this much friction after the peak, when the tire is sliding Seen in previous literature 1 0.8 y F 0.6 0.4 0.2 0 0 5 10 15 20 25 a 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 Stanford University - 29 Dynamic Design Lab Lateral Force with Peak and Slide Friction ms=0.8 and mp=1.6 1 Peak in curve 0.8 0.6 p p0 0.2 and t /t z tp/t p0 0.4 F/F F/Fz 0 -0.2 -0.4 0 0.5 1 1.5 C qtan a 2 2.5 a Can we predict friction on road? Stanford University m p Fz - 30 Dynamic Design Lab 3 Testing at Moffett Field Stanford University - 31 Dynamic Design Lab How Early Can We Estimate Friction? Tire Curve Front slip angle 8000 GPS NL Observer 0.2 0.1 0 0 2 4 6 8 10 12 14 16 ar (rad) Rear slip angle 0.1 0.05 -Lateral Front Tire Force Fyf (N) af (rad) 0.3 7000 6000 5000 4000 3000 2000 1000 0 0 0 2 4 6 8 10 12 14 16 Time (s) linear Stanford University 0 0.05 0.1 0.15 0.2 0.25 0.3 Slip angle af (rad) nonlinear loss of control - 32 Dynamic Design Lab Ramp: Friction Estimates Friction estimated about halfway to the peak – very early! Friction coefficient Steering Angle 2 1.8 1.6 0 2 4 6 8 10 12 14 16 af (rad) Front Slip Angle 0.3 0.2 0.1 0 2 4 6 8 10 12 14 16 1.2 1 0.8 0.6 Lateral Acceleration ay (g) 1.4 Estimated m (rad) 0 -0.1 -0.2 -0.3 0 0.4 -0.5 0.2 -1 0 0 2 4 6 8 10 12 14 16 0 2 4 8 10 12 14 16 Time (s) Time (s) linear Stanford University 6 nonlinear - 33 loss of control Dynamic Design Lab 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 Stanford University - 34 Dynamic Design Lab 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 – our input Model Get slip angles, then tire forces, then derivatives af Stanford University a b b ar V r - 35 Dynamic Design Lab Calculate Slip Angles af a b b ar V r V cos b V sin b ar af V cos b V sin b br V sin b ar V cos b a a f b r V tan a f Stanford University ar V sin b br V cos b b ar b r V tan a r - 36 Dynamic Design Lab Vehicle Model Get forces from slip angles (we already did this) Vehicle Dynamics Fy ma y Fyf Fyr mV ( b r ) z I z r aFyf bFyr I z r This is a pair of first order differential equations Calculate slip angles from V, r, and b Calculate front and rear forces from slip angles Calculate changes in r and b Stanford University - 37 Dynamic Design Lab Making Sense of Yaw Rate and Sideslip r / rad/s 0.4 0.2 0 0 2 4 t/s 6 8 4 t/s 6 8 b / rad 0 -0.1 -0.2 -0.3 0 actual desired 2 What is happening with this car? Stanford University - 38 Dynamic Design Lab For Normal Driving, Things Simplify Slip angles generate lateral forces a Fy Simple, linear tire model (no spin-outs possible) Fyf Caf a f Fyr Cara r Stanford University a Fyf Caf b r V b Fyr Car b r V - 39 Dynamic Design Lab Two Linear Ordinary Differential Equations Fyf Fyr mV ( b r ) aFyf bFyr I z r Caf Car b mV r aC bC ar af Iz Stanford University a Fyf Caf b r V b Fyr Car b r V Caf 1 2 b mV mV 2 2 a Caf b Car r Caf I I zV z aCaf bCar - 40 Dynamic Design Lab Conclusions Engineers really can change the world Many of these changes start with Calculus In our case, change how cars work 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 Stanford University - 41 Dynamic Design Lab P1 Vehicle Parameters a 1.35 m b 1.15 m m 1724 kg N Caf 90000 rad N Car 138000 rad I z 1100 kg m 2 Stanford University - 42 Dynamic Design Lab