Theat assessment fo avoiding collisions with tuning vehicles Mattias Bännstöm, Eik Coelingh and Jonas Sjöbeg Abstact This pape pesents a method fo estimating how the dive of a vehicle can use steeing, baking o acceleation to avoid a collision with a moving object. In the method, the motion of the object can be descibed with an abitay motion model and polygons ae used to descibe its expected extension. The key idea is to paameteize the motion of the vehicle such that an analytical solution can be deived fo estimating the set of manoeuves that the dive can use to avoid the object at discete times. The union of the solutions fo all times is used to estimate how a collision can be avoided duing the complete pediction hoizon. Additionally, a decision-making algoithm is poposed that decides when to initiate autonomous baking to avoid o mitigate a potential collision. A collision avoidance by baking system, based on the poposed method and algoithm, has been evaluated on simued taffic scenaios at intesections. It is shown that a vehicle equipped with such a system can potentially educe the impact velocity with up to 40 km/h in left tun acoss path situations. I. INTRODUCTION The Wold Health Oganization estimates that taffic accidents annually cause 1. million fatalities and as many as 50 million injuies. It is pedicted that these figues will incease by 65% ove the next 0 yeas unless thee is new commitment to pevention [1]. Accident eseach has shown that some accidents ae caused by vehicle failues, but most accidents ae caused by human eos []. One way to educe the numbe of accidents and thei consequences is to actively assist oad uses in thei diving task. This is called active safety o in moe geneal tems peventive safety, anging fom electonic stability contol to dowsiness detection systems. An impotant subset of peventive safety is collision avoidance systems, aiming at assisting the dive in avoiding collisions with e.g. vehicles, bicyclists o pedestians [3]. Seveal vehicle manufactues now offe functionality that help the dive in avoiding collisions by poviding wanings, bake suppot o even automatic intevention baking. These technologies ae often applied to ea-end collision situations. One eason behind this is that ea-end collisions ae common accident scenaios which ae geneally estimated to account fo 3% of all police-epoted accidents and 5% of all fatalities [4]. Anothe eason is that the kinematics of ea-end collisions ae eively easy to pedict, making theat assessment and decision-making moe staightfowad. Theat assessment is commonly used denotation fo algoithms that estimate how oad uses can act to avoid a potential collision. State-of-the-at collision avoidance technology elies on invehicle sensos like ada and camea that constantly monito M. Bännstöm and E. Coelingh ae with the Depatment of Vehicle Dynamics and Active Safety, Volvo Ca Copoation, 40531 Götebog, Sweden mbanns3@volvocas.com, ecoeling@volvocas.com J. Sjöbeg and M. Bännstöm ae with the Depatment of Signals and Systems, Chalmes Univesity of Technology, 4196 Götebog, Sweden jonas.sjobeg@chalmes.se the aea in font of the ca. If the ca appoaches anothe vehicle and the dive needs to undetake an action to avoid an accident, an audible and/o visible waning is povided in ode to incease the dive s attention in such a way that he o she can avoid o mitigate the accident [5]. If the dive does not eact to the waning and it is judged that the dive is unable to avoid a collision without assistance, autonomous baking is applied to mitigate o avoid the potential collision. To futhe incease the scope of collision avoidance technology this pape focuses on theat assessment fo avoiding o mitigating collisions at intesections and oundabouts. Compaed to ea-end collision the diffeences ae: In ea-end collisions only the ea-side of the lead vehicle is of inteest and this side is always appoached at a 90 degees angle. The oientation of a vehicle enteing an intesection may change ove time and thus, the font, ea and side of the vehicle has to be taken into account. The dive of the lead vehicle has limited possibilities to detect the stiking vehicle. The dive of a cossing vehicle has bette possibilities to detect potential theats. Collisions at intesections account fo 41% of all policeepoted cashes, 46% of all injuies and 1% of all fatalities [6,7]. Seveal algoithms fo assessing taffic situations at intesections has been poposed in pevious eseach. Some theat assessment algoithms ae esticted to staight cossing path scenaios, whee deteministic constant acceleation models ae used to descibe the motion of the cossing vehicle [8]. The simplified motion model makes is possible to deive an analytical solution. Algoithms that can assess moe geneal taffic scenaios often use a bute foe appoach, e.g. Monte- Calo simuions [9], to find potential avoiding manoeuves. The analytical algoithms can easily be implemented in ealtime applications, while the bute foce algoithms ae much moe computationally demanding. II. OUTLINE The key idea of the method pesented in this pape is to discetize a multi-dimensional theat assessment poblem into seveal smalle poblems which can be solved analytically. Using analytical solutions makes the method a computationally efficient altenative fo assessing complex taffic scenaios, e.g. involving tuning vehicles at intesections o oundabouts. In Section III, a decision-making algoithm is poposed detemining when to initiate autonomous baking. Decisionmaking algoithms fo autonomous steeing o acceleation ae not consideed in this pape. Section IV contains the poposed method fo estimating if the dive of a vehicle can use steeing, baking o acceleation to avoid a collision with a moving object. In Section V, a collision avoidance by baking system, based on the poposed method and algoithm,
is evaluated on simued taffic scenaios. Finally, conclusions ae dawn in Section VI. III. DECISION MAKING FOR AUTONOMOUS BRAKING Assume that the dive of a vehicle needs to avoid a collision with an object duing a limited pediction hoizon, t [0,T max ]. Denote the vehicle as the host vehicle, the object as a taget and assume that thee ae no othe obstacles pesent. Assume that the dive of the host vehicle o the dive of the taget vehicle can attempt to avoid a collision by eithe baking, steeing o acceleation. Potential coodinated avoidance manoeuves, whee both the host vehicle and the taget vehicle pefoms synchonized avoiding manoeuves ae not assessed. No ae potential combined baking and steeing manoeuves. Although these assumptions may appea too limiting to be useful in pactice, they ae quite easonable when assessing citical taffic situations. If the pediction hoizon is sufficiently small, it is easonable to neglect the possibility to initiate coodinated avoidance manoeuves. Instead of consideing coodinated avoidance manoeuves, it is poposed that the taffic situation shall be assessed both fom the host vehicle pespective and the taget pespective. Fist, the set of host vehicle manoeuves that can be used to avoid a collision is estimated, unde the assumption that the futue tajectoy of the taget is given by the cuent state of the taget. Secondly, the set of taget manoeuves that can be used to avoid a collision is estimated, unde the assumption that the futue tajectoy of the host vehicle is deteministic and given by the cuent state of the host vehicle. Like ealie developed methods [3], inteventions ae inhibited if the dive of the host vehicle has the oppotunity to avoid a collision. The decision-making algoithm pesented in this pape also inhibits inteventions if the taget has the oppotunity to avoid a collision. This is a consevative appoach which educes the isk of unnecessay inteventions, especially at intesections whee thee is a possibility that the dive of a cossing vehicle detects the theat and makes a e avoiding manoeuve. In geneal, it is easie fo eithe of the dive of the host vehicle o the dive of the taget vehicle to pefom a e avoiding manoeuve. Fo example, assume that the host vehicle is appoaching a stationay taget in a ea-end collision situation. Then, it is always easie fo the dive of the host vehicle to pefom an avoiding manoeuve than the othe way aound. Autonomous baking is poposed to be initiated when: The dive of the taget vehicle can not avoid a collision by steeing, baking o acceleating, AND The dive of the host vehicle can not avoid a collision by steeing o acceleating, AND Full baking is needed in the host vehicle to avoid a collision. This algoithm assues that if both vehicles ae equipped with the simila algoithms, potential inteventions in the vehicles will not intefee with each othe in e.g. cossing taffic situations. The autonomous baking is inteupted when the host vehicle no longe need to bake to avoid a collision. IV. THREAT ASSESSMENT In this section, a method is poposed fo estimating how the dive of a vehicle can use steeing, baking o acceleation to avoid a collision duing a limited pediction hoizon. The method is used to deive a theat assessment algoithm. Fo simplicity, the algoithm is descibed the host vehicle pespective, but it can easily be modified to assess the situation fom a taget vehicle pespective. In Section IV-A, the motion model fo the host vehicle is descibed and needed assumptions ae made. Section IV-B contains the method fo estimating how the dive of the vehicle can use steeing to avoid a collision. In Section IV-C, the method is extended to estimate how the dive can bake o acceleate to avoid a collision. Assume that the host vehicle has access to good estimates of the taget state, e.g. its dimensions, velocity, acceleation, yaw ate and oientation. These estimates can be obtained though a good in-vehicle senso fusion system, possibly combined with vehicle to vehicle communication. The poposed method has the following chaacteistics: Abitay motion models can be used fo descibing the motion of the taget. This ensues that it potentially can be applied to ea-end collisions, collisions with tuning objects, but also othe collision scenaios. The taget is epesented by a polygon with abitay shape and numbe of vetices. These can epesent diffeent oad uses such as passenge vehicles, tucks, tucktaile combinations as well as pedestians and bicyclists. The polygonal shape of the taget is allowed to change ove time. This is an impotant chaacteistic when e.g. a tuning tuck-taile combination is to be epesented. The vehicle dynamics of the host vehicle is descibed by a so-called bicycle model [10], such that vehicle slip can be taken into account when judging the possibilities fo collision avoidance by steeing. The bake system dynamics of the host vehicle ae taken into account to be able to ealistically judge to possibilities fo collision avoidance by baking. A. The vehicle model When using the poposed theat assessment method descibed in Section IV-B and IV-C, the motion of the host vehicle has to be paameteized such that only one solution exist fo taveling to a cetain location in a cetain time. One paametization is used fo assessing the possibility to avoid a collision by steeing and anothe paametization is used fo assessing collision avoidance by baking/acceleation. By using the selected paameteizations, the collision avoidance poblem can be solved analytically. In this pape, a bicycle model has been selected to descibe the motion of the host vehicle, as illustated in Fig. 1. Let the oigin of a gound-fixed catesian coodinate system x,y) be positioned at the initial position of the font cente of
unde the assumption that α is small and R R in Fig. 1. Using ) and 5) gives L v ML f L f + L )C v = kv 6) whee k 0 is a constant that only depends on the vehicle weight distibution and the ea coneing stiffness. The appoximation 6) can be used fo both ovesteeed and undesteeed vehicles duing steady-state coneing. Note that the distance L v can exceed the length of the vehicle duing nomal diving conditions and that the appoximation of L v is independent of the adius of tun R. Fig. 1. The motion of the host vehicle is descibed with a bicycle model. the host vehicle. The coodinate system is diected along the cente of the initial oientation of the host vehicle. The length and width of the host vehicle ae indicated by L h and W h. The cente of gavity is denoted as CoG. The distance fom the ea axle to CoG is given by L. The distance fom the font axle to CoG is given by L f. The distance fom the font of the vehicle to the ea axle is L 0. The ea and font slip angles ae given by α and α f, espectively. The steeing wheel angle can be appoximated with δ = α α f + L f + L R whee R is the adius of the tun. The ea slip angle is given by α ML f v ) L f + L )C R whee M is the host vehicle mass and C is the ea coneing stiffness. The font slip angle is given by ML v α f = L f + L )C f R whee C f is the font coneing stiffness [10]. When tuning, the dynamics is modeled with a time delay t d afte which steady-state coneing is obtained instantaneously. Steady-state coneing, means that the tun cente of the host vehicle, x,y ), does not change ove time and the steeing angle δ is kept constant. The initial time delay can be used to compensate fo the tansient behavio until steady-state coneing is achieved, e.g. t d = 0.3s. Time delays ae easy to include in the algoithm and will be left out in the deivation to make the pape easie to follow. The x-coodinate of the tun cente is given by whee 1) 3) x L v L 0 4) L v = R sinα Rα 5) B. Stee to avoid In this section, a method and an algoithm is poposed fo estimating how the dive of the host vehicle can use steeing to avoid a collision with a taget vehicle. The method consist of these steps: 1) Pedict the motion of the taget vehicle with an abitay motion model duing a limited pediction hoizon, t [0,T max ]. ) Divide the pediction hoizon into a seies of discete time steps t i = it s, e.g. t s = 0.05s whee i 1,,..,N and t s = T max /N, as illustated in Fig.. 3) Let the taget be epesented by a polygon of abitay shape and numbe of vetices. The positions of the vetices at time t i ae given by x i,y i ). The shape of the polygon can be changed at each time step. 4) Fo each time step, t i, find all vetices in x i,y i ) that the host vehicle has to avoid and denote them as x i,y i ). 5) Fo each time step, t i, estimate how the dive of the host vehicle can stee to clea all selected vetices, x i,y i ), eithe to the left o to the ight. Both the font end and the ea end of the host vehicle shall clea all vetices, as illustated in Fig. 3-4. 6) The union of the solutions in step 5 is used to estimate how the dive can stee to clea all selected vetices duing the entie pediction hoizon. Step 1-3 ae staightfowad and needs no futhe explanation. Step 4: Only vetices that potentially could be eached at time t i has to be avoided. Assume that the host vehicle speed, v, is constant while the host vehicle is steeing to avoid a collision. Thus, all vetices whee vt i L h < x i < vt i ae eachable at time t i. Denote such vetices as x i,y i ). Step 5: Potential steeing manoeuves ae paameteized such that an analytical solution can be deived fo estimating the manoeuve needed fo avoiding a single cone x i,y i ) x i,y i ). When using a bicycle model with steady-state coneing, the solution is given geometically. A study of the Figues 3 and 4 gives that To pass the cone x i,y i ) on the left side with the font end of the host vehicle, the tun cente is given by y left,font x i x ) + y i x W h 4 y i +W h 7)
Fig.. Example: The host vehicle, illustated at t = 0, is appoaching a tuning taget. The futue position of the taget is pedicted in discete time steps, t i, duing a limited pediction hoizon t [0,T max ]. The poposed method is used to find the set of steeing, baking and acceleation manoeuves that the host vehicle can use to avoid a collision duing the entie pediction hoizon. To pass a cone x i,y i ) on left hand side with the ea end of the host vehicle, the tun cente is given by Fig. 3. The host vehicle position at time t = 0 is illustated along with the polygon epesenting the pedicted position of the taget vehicle at a discete time t i. The figue illustates the steeing manoeuve that the dive of the host vehicle can use to pass a cone of the object, x i,y i ), with the font end of the host vehicle, at a discete time instance t i. x i x ) + y y left,ea i W h 4 8) y i +W h Similaly, to pass the cone on the ight hand side, the tun cente is given by and x i x ) + y y ight,font i x W h 4 9) y i W h x i x ) + y y ight,ea i W h 4 10) y i W h Stoe the solutions fo all time steps and all selected vetices in vectos y left,font, y left,ea, y ight,font and y ight,ea. Step 6: To pass all eachable vetices, x i,y i ) i 1,,..,N, on left hand side, the tun cente is given by [ ] ) 1 1 y,left = max y left,font y left,ea 11) To pass on the ight hand side, the tun cente is given by [ ] ) 1 1 y,ight = min y ight,font y ight,ea 1) The avoiding manoeuve is assessed as feasible if the coesponding eal acceleation does not exceed the maximum allowed eal acceleation, i.e. a a max. The eal acceleation to pass the taget to the left is given by a left = v /R left, and to the ight a ight = v /R ight, whee R left = signy,left ) y,left +L L v ) 13) Fig. 4. The host vehicle position at time t = 0 is illustated along with the polygon epesenting the pedicted position of the taget vehicle at a discete time t i. The figue illustates the steeing manoeuve that the dive of the host vehicle can use to pass a cone of the object, x i,y i ), with the ea end of the host vehicle. Note that anothe time instance is illustated in this figue than in Fig. 3. Compae with the illustation in Fig.. R ight = signy,ight ) y,ight +L L v ) 14) It is judged that the dive can stee to avoid a collision if a left a max OR a ight a max 15) Futhemoe, the steeing angle, given by 1), shall not exceed the maximum steeing angle of the vehicle, δ δ max. C. Bake o acceleate to avoid In this section, a method is poposed fo estimating how the dive of the host vehicle can use baking o acceleation to avoid a collision. While baking o acceleating to avoid a collision, it is assumed that the tun cente, x,y ), does not
change. Similaly to the stee to avoid method in Section IV-B, the method consist of these steps: 1) Pedict the motion of the taget vehicle with an abitay motion model duing a limited pediction hoizon, t [0,T max ]. ) Divide the pediction hoizon into a seies of discete time steps t i = it s, e.g. t s = 0.05s whee i 1,,..,N and t s = T max /N. 3) Let the taget be epesented by a polygon of abitay shape and numbe of vetices. The shape of the polygon can be changed at each time step. Denote the endpoints of all edges of the polygon as x 1,i,y 1,i ) and x,i,y,i ). 4) Fo each time step, t i, find all edges of the polygon that the host vehicle has to avoid. Denote the endpoints of these edges with x 1,i,y 1,i ) and x,i,y,i ). 5) Fo each time step, t i, estimate how acceleation o baking can be used to avoid a collision with all edges selected in step 4. 6) The union of the solutions in step 5 is used to estimate how the dive can bake o acceleate to avoid all edges duing the entie pediction hoizon. Step 1-3 ae staightfowad and needs no futhe explanation. Step 4: Only edges that potentially can be eached by baking o acceleation has to be consideed. Reachable edges fulfill miny 1,i,y,i ) W h AND maxy 1,i,y,i ) W h 16) and ae denoted as [x 1,i,y 1,i ),x,i,y,i )]. Step 5: Denote a single edge at time t i as [x 1,y 1 ),x,y )], whee [x 1,y 1 ),x,y )] [x 1,i,y 1,i ),x,i,y,i )]. To avoid a collision with the edge, the host vehicle has to avoid the x- position inteval [x i,x + i + L h ] at the time t i, as illustated in Fig. 5. The inteval is given by ) x i = x 1 + min x x 1 ) yleft y 1,x x 1 ) yight y 1 17) y y 1 y y 1 ) x + i = x 1 +max x x 1 ) yleft y 1,x x 1 ) yight y 1 18) y y 1 y y 1 whee ) y left Wh = min,maxy 1,y ) y ight = max W ) h,miny 1,y ) 19) 0) In ode to avoid collisions whee the host vehicle comes to a est and then is hit fom the side o font, let x i = minx i,x i+1 ) i 1,,...,N 1 1) If min x ) i 0, a collision can not be avoided by baking. Let the host vehicle acceleation be descibed by an acceleation pofile with one degee of feedom, e.g. { a0 + j at) = t if t [0,t j ] ) a if t > t j as illustated in Fig.6, whee a 0 is the initial host vehicle acceleation and a a 0 + j t j the final acceleation. The Fig. 5. The host vehicle is illustated as time t = 0 and the the pedicted position of the taget vehicle at time t i. The polygonal shape of the taget is divided into edges, whee evey edge has to be avoided. One edge, [x 1,y 1 ),x,y )], is illustated in the figue along with y left and y ight fo the edge and the distances x i and x i fo the entie polygon. acceleation ate j is a vaiable which gives the acceleation pofile one degee of feedom. A suitable selection of t j is t j = max 0, a ) min a 0 3) j min whee a min is the maximum deceleation of the host vehicle and j min is the maximum deceleation ate. Acceleation [m/s ] 0 a 0 5 10 j t j a 0 0. 0.4 0.6 0.8 Time [s] Fig. 6. Example of an acceleation pofile ), whee the initial acceleation, a 0 = m/s, t j = 0.4s, a 10m/s and j 0m/s 3. Since the acceleation pofile only has one degee of feedom, thee is only one way to tavel a cetain distance in a given time. The deivation of the vaiable j fo taveling a distance, x i, in a given time, t i, is given by t j x i v 0 t j a 0 j t t i t j )v+a 0 t i t j ) j ) a 0 t 3 j 6 +t i t j ) t j + t t i t j ) 4) j whee t j = mint j,t i ). To avoid the edge, [x 1,y 1 ),x,y )], by acceleating, put x i = x + i +L h in 4) and denote the solution as j +. The needed acceleation is then given by a + a 0 + j + t j. To avoid the edge by baking, put x i = x i in 4) and denote the solution as j. The needed deceleation is then given by a a 0 + j t j. Stoe the solutions fo all time steps and all selected edges in two vectos, a + and a.
Step 6: To avoid all edges duing the entie pediction hoizon, the dive of the host vehicle must deceleate at least a bake = mina ) o acceleate at least a acceleate = maxa + ). It is judged that the dive can avoid a collision by baking if a bake a min and by acceleating if a acceleate a max, whee a max is the maximum acceleation of the host vehicle, e.g. a max = 4m/s. V. RESULTS FROM SIMULATIONS Let a collision avoidance system CA) be based on the decision-making algoithm in Section III and the method fo theat assessment in Section IV. The CA system has been evaluated on the taffic scenaios descibed below. Let a taget vehicle slow down to 5km/h befoe initiating a 90 o tun with a 1m adius of tun. The taget is acceleating with 1m/s while tuning. The host vehicle dives staight though the intesection at constant speed. The host vehicle speed and initial position ae vaied to ceate diffeent scenaios. Fo both vehicles, assume that a max = 8m/s, δ max = 45 o, a max = 4m/s, a min = 10m/s, j min = 0m/s, t d = 0.3s, W = m, L = 5m, L f = 1.1m, L 1.68m and k = 0.01, as descibed in Section IV. Denote the time to collision as t c. The simuions show that autonomous baking is initiated up to 1s befoe a potential collision. The autonomous baking educes the impact velocity significantly, especially when the initial host vehicle velocity is v [40, 60]km/h. Impact velocity eductions of up to 40km/h ae obseved. Some collisions ae avoided fo host vehicle velocities up to 30km/h. Impact velocity [km/h] Fig. 7. 80 60 40 0 With autobake Without autobake 0 4 0 4 Lateal offset at impact [m] Example of simuions with and without the CA system. VI. CONCLUSIONS In this pape, a novel method has been pesented fo estimating how the dive of a vehicle can manoeuve to avoid a collision with a moving object. In the method, it is assumed that the motion of the object can be descibed with an abitay motion model and that polygons can be used to descibe its extension. The key idea in the method is to estimate how a collision can be avoided at discete times. The solutions fo all times ae joined to obtain an estimate how to avoid a collision duing the complete pediction hoizon. This poblem can be solved analytically fo each time instance, unde the assumption that the motion of the vehicle has one degee of feedom fo steeing and one fo acceleating o baking. As an example, an analytical solution has been pesented fo a bicycle model whee vehicle slip and bake dynamics ae taken a) t c = 1.5s b) t c = 1.0s c) t c = 0.9s d) t c = 0.0s e) t c = 0.3s Fig. 8. The host vehicle bottom) is appoaching the taget vehicle top) at 50km/h. The gay vehicle is equipped with a CA system. The coss on top of the vehicles indicates how the dive of the vehicle can stee, bake o acceleate to avoid a collision. In a), it is easy fo the taget to avoid a collision by baking. In b), it is impossible fo the host vehicle to avoid a collision by baking, steeing o acceleating, while the taget vehicle still can avoid a collision. In c), autonomous baking is initiated in the host vehicle to mitigate the collision. In d), the host vehicle without the CA system collides. In e), the host vehicle with a CA system collides. The duation of the bake intevention is 1.s and the impact velocity is educed with 34km/h. Note that a CA system in the taget vehicle could avoid the collision by baking in b). into account when judging if the dive of the vehicle can avoid a collision. A collision avoidance by baking system, based on the poposed method and a new decision-making algoithm fo autonomous baking, has been evaluated on simued taffic scenaios. The simuions indicate that thee is a high potential fo using autonomous baking to avoid o mitigate collisions at intesections. 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