OPTIMAL VEHICLE MOTION CONTROL TO MITIGATE SECONDARY CRASHES AFTER AN INITIAL IMPACT

Transcription

1 Proceedings o the ASME 4 Dynamic Systems and Control Conerence DSCC4 October -4 4 San Antonio X USA OPIMA VEHICE MOION CONRO O MIIGAE SECONDARY CRASHES AER AN INIIA IMPAC Byung-joo Kim Department o Mechanical Engineering Uniersity o Michigan Ann Arbor MI USA bjuim@umich.edu Huei Peng Department o Mechanical Engineering Uniersity o Michigan Ann Arbor MI USA hpeng@umich.edu ABSRAC ypical driers are not ready to react to unexpected collisions rom other ehicles. he initial impact can startle the drier who then ails to maintain control. Since a loss o control leads to intense sidding and undesirable lateral motions more seere subsequent eents are liely to occur. o reduce the seerity o possible subsequent (secondary) crashes this paper considers both ehicle heading angle and lateral deiation rom the original driing path. he research concept here is dierent rom today s electronic stability control systems in that it actiates the dierential braing een when the magnitude o yaw rate or ehicle slip angle is ery high. In addition the lateral displacement and yaw angle with respect to the road are part o the control objectie. he inear ime Varying Model Predictie Control (V-MPC) method is used with the ey tire nonlinearities captured through linearization. We consider tire orce constraints based on the combined-slip tire model and their dependence on ehicle motion. he computed high-leel (irtual) control signals are realized through a control allocation problem which maps ehicle motion commands to tire braing orces considering constraints. Numerical simulation and analysis results are presented to demonstrate the eectieness o the control algorithm. INRODUCION Approximately.6 million motor ehicle crashes were reported in in the United States resulting in about 33 atalities []. Among them oer 3% o all crashes hae two or more harmul eents ollowing the initial collision. Seeral statistical studies based on ehicle crash data [-4] indicate that number o multiple impact crashes has been increasing. hese reports ound that the ris o seere injuries is much higher in multiple impact crashes than in single impact crashes. In [] it is also pointed out that the ris o both injuries and atalities increased with the number o collision eents. Multiple impact crashes can be more dangerous because driers are not prepared or impacts rom the side or bac as the most driers ocus on the ront iew. Since looing ahead and using an inside reariew mirror cannot always proide enough inormation o adjacent lanes [6] the continuous monitoring o surroundings while maintaining a lane may not be an easy tas or the aerage drier. or this reason driers can panic or ail to respond to an unexpected initial eent and thus lose control. When driers react improperly undesirable ehicle motions can lead to hazardous heading angles and excessie lateral deiation resulting in harmul secondary crashes. One o the crash analysis reports in NHSA (National Highwaaic Saety Adistration) [7] presents the ehicle spin angle distribution in the most harmul secondary eent crashes. It shows that secondary eents at around 9 turns either clocwise or counter clocwise cause the most harmul secondary crashes. As shown in [8] based on the injury scale leel data side impacts in a secondary eent are more harmul to the occupants. Since the sides o a ehicle hae less crashenergy absorbing structures than the rontal and rear sections the ris o atalities and serious injuries with side crashes is higher [9]. Moreoer an analysis inestigating ehicle dynamic motion ater an impact in [] shows that an excessie lateral lane deiation ater an initial impact plays a eole in the ris o secondary impacts. Secondary crashes can be partly mitigated by employing a ehicle Electronic Stability Control (ESC). Howeer the eectieness o ESC to mitigate secondary crashes has not been ully taen adantage o []. Since these control algorithms were designed to interene ater a ehicle-to-ehicle impact only limited control actions can be applied. Some ESC could een turn o i the yaw rate or lateral acceleration is too high because the situation can be misinterpreted as a sensor ailure. Hence a new eature must be designed to mitigate secondary crashes. o enhance ESC perormance ater an initial crash seeral control concepts in the literature were deised using steering or braing [-4]. Recently a secondary Copyright 4 by ASME

2 collision braing strategy has been aailable on production ehicles by the name o Secondary Collision Brae Assist. EURO NCAP (New Car Assessment Program) beliees that this system can reduce atalities and serious injuries []. Howeer these systems are usually either limited to wor ater a mild collision scenario or did not consider hazardous heading inal angle. Considering the act that the sides o ehicles are more ulnerable than the ront or rear the ehicle heading angle is an important parameter that needs to be controlled or secondary collision saety. In addition a small lateral deiation ater an initial collision is desired to reduce the ris o secondary collision with roadside objects or ehicles in other lanes. he proposed control concept in this paper includes allowing a high yaw motion to aoid ulnerable heading angles (e.g. 9 7 ). his paper ocuses on a ehicle motion control strategy which leads a ehicle to a sae heading angle ( 8 36 etc.) while maintaining small lateral displacement. he underlying assumption is that the sensors and actuators still unction normally ater the irst impact. he heading angle and lateral speed are assumed to be aailable rom a reliable ision magnetometer or GPS sensor [6]. And no drier steering action is considered in the current design. VEHICE MODE AND IRE MODE he ehicle model considers longitudinal lateral and yaw motions o the ehicle as shown in ig.. he goerning equations are: mv X x cos cos x r R y sin sin R mv Y x sin sin x r R y cos cos R I r a sin cos R R b yr / cos W x R x W / y sin R y x r R x r zz x y he longitudinal and lateral orces or each tire are denoted by x y which are deined in a tire ixed coordinate. he subscript ( r ) represents the ront and rear axles and the next subscript denotes the let and right side o the ehicle ( R ). x and y are axes o a coordinate system located at the center o graity (CG) in the ehicle ixed rame while X and Y indicate ehicle position in the inertial coordinate. he motion o the ehicle is deined by ehicle elocities VX V Y associated with X and Y the steering angle δ at the ront tires the heading angle ψ and the yaw rate r. Vehicle geometries are represented by ab and W which are distances to ront and rear axles rom the CG location and the width o the ehicle. M is the ehicle mass and Izz is the ehicle yaw moment o inertia. It should be noted that the equations are ormed in inertial coordinate to explicitly consider the lateral IGURE. Vehicle motion with respect to a lane in the XY-plane displacement. Considering the act that longitudinal brae orces o each tire are independent controls in dierential braing Eq. () () and (3) can be constructed as a six-state nonlinear irst-order system: t t g u t where X V Y V r and t g u t X Y VX y sin sin / M R VY y cos cos / M R r ay cosb sin R / I zz W /y sin R y x cos cos / x r M R x sin sin / x r M R W ax sin x r R x r / I zz W /x cos R x y sin sin / M R y cos cos / M R ay cos b R y r R / I zz W /y sin R y (4a) (4b) Copyright 4 by ASME

3 Here u represents the independent 4-wheel brae control actions ( u x x R x r x rr ) y are the lateral tire orces at a nown slip angle under zero longitudinal slip condition and y are the induced lateral tire orce ( y y y should be noted that (u) while g ut ) caused by the combined-slip eect. It t is not related to the control inputs is split into two parts: the direct eect o control inputs ( x ) and indirect eect (disturbance) due to induced lateral orce ( y ). ater we will sole the control problem in two steps: irst sole the irtual control input g(ξut) and then compute the tire orce x that will trac the irtual control input despite o the combined-slip tire nonlinearities. or an eectie ehicle control action accurate tire orce characterization is important. We use Paceja s Magic ormula tire model [7] which describes combined longitudinal and lateral orces as unctions o the tire slip angles (α) the slip ratio (λ) and ertical loads ( ): C s p z x tire y tire P C E s pz s where s s s tan z x y (a) P C EsinCacrtan E arctan C C C (b) 3 z pz z /. M g (c) c z Mg C c M g e (d) In this unction road riction coeicient () is assumed to be ixed on a dry asphalt (.8 ). he tire slip angles are: y ar arctan x / W r y ar R arctan R x / R W r yr br r arctan xr / W r yr br R r arctan R / r xr R W he tire orce proiles oer the entire range o tire slip angle and longitudinal slip ratio are shown in ig.. When α is near or ±8 both longitudinal and lateral orces ary signiicantly with λ. On the contrary the eects o λ are small when α is around ±9 meaning that braing action is not eectie. IGURE. ongitudinal and lateral orces as unctions o the tire slip angle and longitudinal slip ratio (- λ ) ANAYSIS O EECIVE IRE ORCE RANGES Because o the coupling eect o longitudinal-lateral orces the induced lateral tire orce ( y ) generated by the tire brae orce ( x ) needs to be considered to compute the ehicle lateral and yaw motions. Equation ()-(3) show the ehicle yaw moments ( M z ) calculated by the moment arm rom the CG point to the orces x and. y M z sin / cos x a W acos / sin y z R x R W y R W W M asin / cos acos / sin zr xr W yr M / b zr R xr R W yr R M / b rom these inematic relationships possible ranges o yaw moment acting on a ehicle CG point can be obtained with arious wheel slip ratios rom ree rolling (λ=) to ully loced (λ=-). As shown in ig. 3 it is noted that the most eectie wheel to change yaw moment can be detered based on the ehicle side slip angle. or example there might be a case when the ehicle needs a positie yaw moment and the slip angles o all tires are small positie. In this case the ront-let wheel is the best candidate because the yaw moment on that tire M x [N] y [N] ( z =- = [deg] =- = [deg] ) shows a greater positie alue than others. Similarly right rear wheel is the best to use or negatie yaw moment generation under the same situation. It should be noted that there exist regions that hae almost no control authorities or yaw moment control. In ig. 3 it can be seen that when the tire slip angle is around ±9 degrees little yaw moment can be generated. i.e. it is hard to control ehicle yaw motion in those situations. Examples aboe indicate the importance o understanding ehicle dynamics especially tire saturation 3 Copyright 4 by ASME

4 constraints when designing a controller such as the model predictie control method to be used in this paper. Although we only presented the ehicle yaw motion results one also should ind other easible control regions or longitudinal or lateral ehicle motions. In those cases the ehicle heading angle needs to be considered together with the tire slip angle and slip ratio. INEAR IME VARYING MODE PREDICIVE CONRO he irst step to implement the model predictie control (MPC) is the linearization o the nonlinear ehicle dynamics about eery operating point so that a quadratic programg structure can be applied [8 9]. his method decomposes the nonlinear design problem into seeral linear sub-problems. he successie linearization points do not need to be equilibriums. MPC inds a cost-imizing control sequence oer the prediction horizon. At the same time it incorporates easible control bounds so that control signals are implementable critical or the MPC to wor satisactorily. Moreoer the MPC is applicable to real-time processing because o the low computation requirement o the linear time-arying MPC (V- MPV). Architecture o the Proposed Control System he oerall control structure is shown in ig. 4. he desired ehicle states are irst compared with their current states. hen in response to the state errors the V-MPC controller deteres the desired irtual controls based on the QP (Quadratic Programg) optimization solution under easible ehicle dynamic constraints. Coupling between the tire longitudinal and lateral orces is considered in iguring out the control constraints. Next the optimal control allocation process maps the irtual control demand onto indiidual wheel brae orces. In the last stage actuator controllers manipulate physical ariables such as wheel cylinder braing pressures to achiee the desired tire orces. hen this actuator action aects ehicle motion and the resulting ehicle states are measured or estimated or the eedbac control. M z [Nm] M zr - [Nm] [] [] r IGURE 4. Architecture o the proposed control system inearizing at Non-Equilibrium Points Considering the nonlinear dynamics shown in Eq. (4) the linearization is done through aylor expansion around points ( ) and irtual control input ( ) is introduced rom the simple relationship ( g ut B ): t t At B t t A t B t B t t where At B (4a) B I identity matrix (4b) t (4c) Y Mz t t At (4d) Here we exclude the irst two states ( ) and the model is reduced to a 4-state system ( because we are mainly concerned about the ehicle lateral displacement error and yaw directional motion. rom Eq. (4) the numerical [Nm] [] IGURE 3. Vehicle yaw moments that can be generated by braing each wheel. he shaded area between red line (upper bound) and blue dotted line (lower bound) depicts achieable region or all possible slip ratio (- λ ) M z R M zrr - [Nm] [] R r R 4 Copyright 4 by ASME

5 discretization is applied to build a time-arying linear discrete time state-space system with a sampling time ( ): A d B d B d Here we assume that the coeicients in the system and input matrices are constant oer the horizon n : A A A A B d d d n B d B B d n he sequence o prediction state oer the time horizon (nsteps) can be expanded as: A B B A AB B AB B A A B A B n n n n A B n AB n B n n n A B A B AB B Note that the states at a uture moment are dependent on the current states and other sequence terms. his can be summarized with a matrix orm: G H V W n n where A A 3 3 G A V n n n n A n B AB B H AB AB B n n n3 A B A B A B B B AB B W AB AB B n n n3 A B A B A B B (a) (b) (c) Optimal Problem ormulation he control goal is to imize both the lateral deiation rom the original course and to achiee a sae heading angle while imizing control eorts. So the cost unction is deined as the summation o weighted state deiation and weighted control input sequence. n n i i i i i i J Q R where Q and R are diagonal weighting matrices and the state deiations are i i. he desired state ( desired desired ) are predetered by the oline optimal computation. In the present wor the structure o enironment or the controlled ehicle is assumed to be nown as straight road with multiple lanes. So the oline optimal calculation deteres a saer heading angle ( 8 36 etc.) which imizes the lateral deiation under the gien initial impact conditions. Here we set the same time horizon (n) or both predicted states and input sequences. But once the optimal control set * * ( V ) is ound only the irst control step ( ) will be n implemented. hen Eq. () can be rewritten using a more compact notation as ollows: J diag Q V diag RV n n n n where diag Q and diag R are bloc diagonal matrices. By substituting Eq. () one can rearrange Eq. (3) so that the problem becomes a quadratic orm o V : n J V H QH R V V H Q G W n n n G QG W QG W QW Since the last three terms in Eq. (4) are not aected by the input sequence ( V ) those terms can be ignored when n iguring out the optimal input sequence to imize J. his allows the control objectie to be ormulated in a simpler quadratic orm: J V S V V V n n n n where S H QH R H QG W (a) Constraints Handling Constraints on the control magnitude and rate o change can be deined by the ollowing inequalitelationships: n n n ii Note that the control limits ary along the prediction horizon as shown in ig. 3. his is because the ranges between imum and imum irtual controls changes with tire slip angle and heading angle. In other words the constraints are unctions o the states and control inputs. Ater rearranging and combining Eqs. (6) and (7) one can obtain a simpliied constraint expression based on the deriation in [8] as: Copyright 4 by ASME

6 h V n n hen we hae a general quadratic programg problem with inequality constraints which can be soled by using MAAB Optimization oolbox (quadprog). J V V n n subject to h V n n HE OPIMA AOCAION PROBEM he control laws deried in the preious section compute the optimal irtual controls. As shown in ig. 4 the irtual control commands are ed to the wheel brae optimal allocation module to detere the eectie physical controls. Since we assume that the real controls are the indiidual wheel braes the outputs o this module are the longitudinal wheel brae orces o each tire. Speciically the objectie o this module is * 4 to ind optimal control u to achiee the irtual control * sequence. ire orce Relationship In Eq. (4) the relationship rewritten as: g ut B can be g ut Bu uwd d where d y y R y rr o replace the induced lateral orce term as a unction o we utilize the orce coupling eect. he tire orces are constrained by an eneloping cure called a riction circle or riction ellipse []. he Magic ormula used in this research inherently describes combined orces within this riction ellipse and this proile can be constructed as a linear unction with manipulating tire model equations. rom the deinition in Eq. () one can analytically get the gradients o orces ( x ) with respect to (= y s x ). s C s C s Cs xtire p y x P 3/ P s x s x sy p p sx sy ss C s C s Css y tire p x y x y P 3/ P s x s x s p p sx sy y y tire ytire ytire s x y tire sx x tire sx x tire sx x tire 44 his matrix is a diagonal matrix which maps the tire longitudinal orces onto the lateral orces y x y y R y rr diag x x R x r x rr hen Eq. (3) becomes: B B W u u d Solution o the Allocation Problem he optimal allocation problem soles the linear relationship Eq. (3) in the least-squares sense subject to the actuator saturation limit. Since the actuators are wheel brae orces the control bounds are u. Because we x x consider a brae control action only the alue u should be negatie. We employ the ollowing least-squares problem or optimal allocation: u * u d u w B W u B where the parameter w is a positie weighting actor chosen to achiee the relationship in (3) as close as possible. subject to u u u SIMUAION RESUS he simulated scenario is that two ehicles are inoled in a crash on a straight road. hen the collision leads the ehicle to trael into the opposite traic lane i no control is applied. or this simulation all state measurements such as position speed yaw rate and heading angle are assumed to be aailable and accurate and actuator delays are not included. In addition all actuators are assumed to unction normally. he ehicle used in the simulation is a big SUV deined in Carsim the ehicle dynamics sotware ( ). It is assumed that the ehicle is traeling straight with an initial longitudinal speed o 3m/s on a lat and straight road. he irst impact is assumed to result in the initial conditions: lateral speed.m/s heading angle 9.º and yaw rate 4.6º/sec. he discrete time to run simulation is.sec and the sampling time or eery linearization and the time horizon or MPC are set to. sec ( steps). Y[m] Vehicle proile Velocity ector ) No Control ) ull Braing ( =-.9 r =-.6) 3) Yaw rate control(esc) & Braing 4) Proposed Control -8 X[m] IGURE. Vehicle trajectory comparison results or three control strategies. (Vehicle sizes are doubled.) 6 Copyright 4 by ASME

7 4 Y Y[m] 3 No control Proposed control Max.deiation (6.8m) X[m] Mz 4 (a) 3 4 ime [sec] Y[m] 3 Max.deiation (6.7m) IGURE 6. Virtual control inputs rom the proposed control strategy. easible boundaries are shown by gray shaded regions between blac lines. Virtual controls soled by MPC are shown with bold lines. x [N] r xr 3 4 ime [sec] [N] 3 4 ime [sec] [N] 3 4 ime [sec] IGURE 7. ongitudinal tire slip ratio commands and corresponding longitudinal tire orces calculated by the optimal allocation problem. he gray shaded regions depict easible control bounds. Weights on tracing error and input rates are Q= diag[ ] and R= diag[..]. our ehicle trajectories with dierent control strategies are presented in ig. or comparison. It is shown that the ehicle with the proposed controller settles into a sae inal heading angle o 8º and returns to the original lane. In contrast the ehicles without braing interention and with other control strategies depart rom their original lane and can be broadsided by ehicles in other lanes. In the case o yaw rate control the brae control actions to change the ehicle yaw motion are ery limited at the R x R rr xrr [N] 3 4 ime [sec] X[m] (b) IGURE 8. Vehicle trajectories under seeral initial conditions representing dierent leels o impact: (a) yaw rate 8~ /s and heading angle 8~7 (b) yaw rate 6~ /s and heading angle 8~3 the initial lateral speed (m/s) and longitudinal speed (3m/s) are the same. end o the maneuer because the tire slip angles are all around ±9º as analyzed in ig. 3. Moreoer attempting to return the ehicle to the original lane (i.e. heading angle to the original º) can cause a large lateral deiation []. he irtual control bounds and control results rom V-MPC are shown in ig. 6. It should be noted that the easible ranges ary with ehicle states; and the solution rom MPC is reasonable only when considering these realistic constraint conditions. Although the constraints show a nonlinear nature the quadratic programg is still applicable to sole the MPC problem through the system linearization process at eery operation point. he results o optimal allocation are shown in ig. 7. At the beginning o the control process the brae control commands are detered to achiee a rapid yaw rotation. In this case the rear wheel brae reactions are shown to hae little aster responses than the ront ones. his action maes it possible to pass by the region where the eectieness o brae control (control authority) is little or none (around second). Ater that wheel commands are regulated to ollow the yaw moment which leads the ehicle heading angle to the desired state 8º. In ig. 8 the capability o the proposed control strategy is ealuated with seeral tests executed in dierent initial impact conditions. he simulated crash scenario has a similar layout to that o ig. but with dierent alues or initial yaw rate and heading angle representing arious leels o initial impacts. Oerall it is seen that the proposed control reduces the imum lane deiations and brings the ehicle bac to the original lane with the desired heading angle o 8º or 36º. 7 Copyright 4 by ASME

8 Although the 8º heading angle might be considered as a dangerous ehicle state by some the situation is still saer than the case where the ehicle is exposed to a side impact rom approaching traic in the lanes other than the original lane. CONCUSIONS his paper presents a V-MPC and optimal allocation algorithm to mitigate the secondary collision ater an initial impact. hese two optimization ormulations exploit the easible control bounds based on the physical constraints. rom the proposed control system wheel brae commands are detered so that the impact induced ehicle motion can be settled into a sae heading angle with a small lateral deiation. Simulations are perormed or arious initial conditions o yaw and lateral motion due to an impact. Simulation results show that the proposed control algorithm can eectiely lead a ehicle to a desired heading angle with a smaller lateral deiation comparing to results rom the uncontrolled case and other control strategies. o enhance the control eectieness uture wor will add an actie steering control to expand the control authority with changing side slip angles when brae controls are limited. he oerall control objectie can then be achieed by an integrated control strategy. Moreoer the practical perormance o the designed scheme can be urther studied by addressing the eect o drier s steering and braing action. A coordination or prioritization strategy between the control commands and the drier action needs to be considered. REERENCES [] NHSA "raic Saety acts" US Department o ransportation. [] J. Bahouth and K. Digges "Characteristics o Multiple Impact Crashes hat Produce Serious Injuries" in Proceedings o the 9th International echnical Conerence on the Enhanced Saety o Vehicles Washington DC USA. [3] U. Sander K. Mroz O. Boström and R. redrisson "he eect o pre-pretensioning in multiple impact crashes" in st International echnical Conerence on the Enhanced Saety o Vehicles (ESV) 9. [4] A. ogawa D. Muraami H. Saei C. Pal and. Oabe "An Insight into Multiple Impact Crash Statistics to Search or uture Directions o Counter-Approaches" in th International echnical Conerence on the Enhanced Saety o Vehicles (ESV). [] J. Zhou "Actie Saety Measures or Vehicles Inoled in ight Vehicle-to-Vehicle Impacts" Ph.D. hesis he Uniersity o Michigan Ann Arbor MI 9. [6]. Ayres. i D. rachtman and D. Young "Passengerside rear-iew mirrors: drier behaior and saety" International journal o industrial ergonomics ol. 3 pp [7] A. Eigen and W. Najm "Problem deinition or pre-crash sensing adanced restraints" DO HS [8] J. enard and R. rampton "wo-impact Crashes- Implications or Occupant Protection echnologies" in Proceedings o 8th International echnical Conerence on the Enhanced Saety o Vehicles Nagoya Japan 3. [9] B. ildes J. C. ane J. enard and A. Vulcan "Passenger cars and occupant injury: side impact crashes" 994. [] D. Yang B. Jacobson and M. idberg "Beneit prediction o passenger car post impact stability control based on accident statistics and ehicle dynamics simulations" in Proceedings o st IAVSD Symposium on Dynamics o Vehicles on Roads and racs 9. [] S. A. erguson "he eectieness o electronic stability control in reducing real-world crashes: a literature reiew" raic Injury Preention ol. 8 pp [] C. Chan and H. an easibility Analysis o Steering Control as a Drier-Assistance unction in Collision Situations IEEE rans. On Intelligent ransportation Systems pp. 9. [3] D. Yang. Gordon et al. Post-Impact Vehicle Path Control by Optimization o Indiidual Wheel Braing Sequences Proc. o th International Symposium on AVEC pp [4] J. Zhou J. u and H. Peng Vehicle stabilization in response to exogenous impulsie disturbances to the ehicle body In Proc. o the American Control Conerence St. ouis MO pp [] EURO-NCAP. (). Audi Secondary Collision Brae Assist. Aailable: audi_secondary_collision_brae_assist.aspx [6] J. Yoon H. Peng Sideslip Angle Estimation Based on GPS and Magnetometer Measurements Proc. th International Symposium on AVEC Seoul Korea. [7] H. Paceja ire and Vehicle Dynamics (Second edition) SAE International. [8] J. Maciejowsi Predictie Control with Constraints. Prentice-Hall ondon [9] P. alcone. Borrelli J. Asgari H. seng and D. Hroat Predictie actie steering control or autonomous ehicle systems IEEE rans. Contr. Systems echnology ol. no. 3 pp [] J. Wong heory o Ground Vehicles 3rd Edition. John Wiley & Sons. [] J. Zhou J. u and H. Peng Vehicle Dynamics in Response to the Maneuer o Precision Immobilization echnique In Proc. o ASME Dynamic Systems and Control Conerence Ann Arbor MI 8. 8 Copyright 4 by ASME