Detection of Dangerous Cornering in GNSS Data Driven Insurance Telematics

Insurance telematics: Opportunities and challenges with the smartphone solution Driving style recognition using a smartphone as a sensor platform
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Published Date:18-12-2017
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1 Detection of Dangerous Cornering in GNSS Data Driven Insurance Telematics Johan Wahlstrom, ¨ Isaac Skog, Member, IEEE, and Peter Handel, ¨ Senior Member, IEEE Abstract—We propose a framework for the detection of dan- gerous vehicle cornering events, based on statistics related to the 13:37 85% no-sliding and no-rollover conditions. The input variables are estimated using an unscented Kalman filter applied to global navigation satellite system (GNSS) measurements of position, speed, and bearing. The resulting test statistic is evaluated in a field study where three smartphones are used as measurement probes. A general framework for performance evaluation and estimator calibration is presented as depending on a generic loss function. Further, we introduce loss functions designed for applications aiming to either minimize the number of missed detections and false alarms, or to estimate the risk level in each cornering event. Finally, performance characteristics of the estimator is presented as depending on the detection threshold, and on design parameters describing the driving behavior. Since the estimation only uses GNSS measurements, the framework is particularly well-suited for smartphone-based insurance telemat- Fig. 1. Example of smartphone display used as an extension of the vehicle’s ics applications, aiming to avoid the logistic and monetary costs dashboard. The displayed real-time feedback includes indicators of speeding, associated with e.g., on-board-diagnostics or black-box dependent driving smoothness, acceleration, swerving, braking, and cornering. solutions. The design of the estimation algorithm allows for instant feedback to be given to the driver, and hence, supports the insurer. While some commercial UBI programs are avail- the inclusion of real time value added services in usage-based- able on the market today, they are mainly based on information insurance programs. extracted from the cars’ on-board-diagnostics (OBD) system, IndexTerms—UBI, Insurance telematics, GNSS, Vehicle lateral or from externally installed hardware components, referred forces, Unscented Kalman filtering. to as black-boxes or aftermarket devices. The US market leader, Progressive Casualty Insurance, initialized their first UBI-program in the late 1990s, and today has approximately I. INTRODUCTION one and a half million policyholders signed up to their current Frameworks for the detection of dangerous vehicle corner- UBI-program, Snapshot 3. After a 30-day trial period where ing events are foremost motivated by safety aspects. Despite speed and time data are collected from the OBD system, each efforts to stabilize rollover inclined vehicles, e.g., sport utility driver is offered to sign up for a vehicle insurance with a vehicles 1, and efforts to improve the conditions of the road premium discount based on the collected data. The discount surface and the quality of the tires, skidding and rollover is calculated based on the total elapsed distance, how often events still play a major role in many of today’s car accidents. the driver makes sudden stops, and the time of day when the Moreover, statistics show that even though only three percent data was recorded. of all vehicle crashes involve a rollover, approximately one- Currently, the commercial expansion of the UBI industry third of all passenger deaths are related to rollover events 2. is held up by the process of acquiring data, which involves As of yet, no safety system exists that can fully compensate for large costs related to installation, maintenance, and logistics. the dangers in turning events induced by excessive speeds or The use of smartphones for the collection of driving data reckless driving. However, the industry forecasts, predicting has been identified as a promising alternative, due to the a paradigm shift as a result of the growth of usage-based- high penetration of smartphones among end-users, and the insurance, assert that all drivers one day will be given the efficiency of wireless data transfer. Approximately one billion option to be financially compensated for safe driving. smartphones were sold in 2013, and for the first time in A usage-based-insurance (UBI) is an automobile insurance history the number of sold smartphones exceeded the number where the insurer uses data on driving behavior to set the pre- of sold feature phones, i.e., mobile phones only providing mium offered to each policyholder. The premiums are adjusted basic telephony. Recent estimates predict that up to 30% of so as to reflect the individual driver risk profiles constructed by all vehicles in the United States, and 60% of all vehicles in the United Kingdom, will be insured through some type of J. Wahlstrom, ¨ I. Skog, and P. Handel ¨ are with the ACCESS Linnaeus Center, insurance telematics (UBI by means of telecommunication) Dept. of Signal Processing, KTH Royal Institute of Technology, Stockholm, program by the year 2020 4. Sweden. (e-mail: When operating as a measurement probe, the smartphone2 is typically mounted in the windshield (see Fig. 1), where estimated attitude. The estimate is however sensitive to both it can provide the driver with information regarding both his GNSS outages and the chosen initialization method. Typically, driving and the surrounding traffic. (Note, however, that the it is assumed that the initial state vector can be estimated dur- smartphone of course is allowed to be picked up and moved ing a period of zero acceleration 23, which severely limits the around inside of the car.) With the use of the collected data, utility of the estimation framework in telematics applications. the insurer can individually tailor the real-time feedback given Since basic smartphones usually do not provide raw GNSS to each driver, and reward low-risk policyholders by offering measurements (pseudorange, carrier-phase, or doppler shift them a premium discount. Commonly employed figure of estimates), GNSS-aided INSs are limited to have a loosely merits (FoMs) in the construction of the drivers’ risk profiles coupled system architecture 24. Moreover, the estimation is include measures based on speeding, driving smoothness, complicated by the fact that the smartphone is not required harsh acceleration, harsh braking, swerving, eco-driving, and to be fixed inside of the car, which means that movements harsh cornering 5. The use of smartphone sensors for risk of the smartphone relative to the vehicle must be identified assessment of driving behavior has recently been studied in and separated from the dynamics of the vehicle itself. Due 6–9. to the increased update rate and the need to estimate both The collected driving data can also be used in complemen- sensor bias and the attitudes of the vehicle and the smartphone, tary projects with aims of societal value, such as reducing the computational cost of an AHRS or a GNSS-aided INS congestion and emissions 10. Several studies has proposed would presumably be several orders of magnitude larger than the fusion of telematics data with data from vehicle detection an implementation only navigating on GNSS measurements. loops (detecting vehicles passing or arriving at some fixed The object of this paper is to present and study the accuracy point) when constructing e.g., incident detectors, or estimators of a method for the detection of dangerous vehicle cornering of expected travel times 11–13. events, solely based on GNSS data. (Related studies utilizing Prevailing challenges currently limiting the expansion of the measurements from e.g., inertial measurement units (IMUs) insurance telematics industry include privacy considerations or optical sensors measuring the steering angle, can be found for end-users 14, 15, the implementation and design of in 25–27.) The paper is organized as follows: Section attractive value added services 10, the storage of data 16, II derives statistics related to the no-sliding and no-rollover and the issue of correlating driving behavior with drivers’ conditions, and use these to define a cornering event in terms insurance claim history. Furthermore, the global navigation of vehicle dynamics. Section III then discusses the method satellite system (GNSS) receivers commonly installed in employed to estimate the vehicle dynamics. A framework for smartphones are often of low quality, and the high presence performance evaluation and mapping between estimated and of errors in the data demands a substantial use of data true cornering events is provided in Section IV, while the processing algorithms to increase the reliability of the data. results of the conducted field study are presented in Section Although motion sensors such as accelerometers, gyroscopes, V. Some final conclusions are drawn in Section VI. and magnetometers, now are included in most smartphones, their potential use in telematics applications is constrained by II. PROBLEM FORMULATION several factors. As an example, note that high rate velocity estimates based on integration of accelerometer measurements In this section, we will present the test statistic which is used requires an accurate estimate of the smartphone’s orientation to determine whether the driver did engage in a (dangerous) 17. However, low-cost gyroscopes embedded in smartphones cornering event at time t . We begin by examining under n exhibit a great deal of noise, and will in a short period of what conditions skidding and rollover events occur. For both time induce a noticeable drift in the estimated orientation. types of events, we will assume that the vehicle’s pitch and The estimate can be stabilized in e.g., an attitude and heading roll angles are equal to zero, and that the driving trajectory reference system (AHRS) 18, 19, which corrects the gy- can be locally approximated with a circle. Values of the roscope predictions by using low-pass filtered accelerometer generic variable c are by notation separated as measured c , and magnetometer measurements to estimate the direction of estimatedc , developing in continuous timec(t), or developing the gravity vector and the magnetic north. The accuracy of in discrete time c , where n is the index of the sampling n the estimated orientation is degraded by, among other things, instance t . n vibrations of the vehicle engine 20, accelerations of the First, we derive the no-sliding condition using arguments car, and magnetic disturbances 21. Since the AHRS updates from classical mechanics. To this end, note that the force requires measurements performed over a period long enough to of friction F exerted on the tires, is limited by the friction f eliminate the influence of most high frequency errors, but short equation enough to justify the assumption that the estimated quantities ? F F (1) f are constant, performance typically drops during high dynamic movements. which says that the force of friction can never exceed , Another option is to employ a GNSS-aided inertial navi- the coefficient of friction, multiplied by the normal force ? gation system (INS) where the errors of velocity estimates, F , exchanged between the tires and the road surface. The propagated using measurements from high rate sensors, are coefficient of friction is uniquely determined by the vehicle’s bounded by continuous updates from GNSS measurements speed, the properties of the tires, and the road surface. The 19, 22. This will generally increase the bandwidth of the normal force will be equal to the force of gravity on the car,3 TABLE I TYPICAL VALUES OF THE TIRE FRICTION COEFFICIENT 32. ? F =mg ma n Speed Tire condition Water depth on road surface km=h 0 mm 1 mm 2 mm p 2 2 2 The coefficient of friction:  F =m v +a h n n n mv n n 50 new 0:85 0:55 0:5 y 50 worn 1 0:4 0:25 90 new 0:8 0:3 0:05 y 90 worn 0:95 0:1 0:05 y Tire tread depth of 1:6mm. ? Fig. 2. To avoid a sliding event, the normal force F , multiplied by the coefficient of friction , must exceed the horizontal force F . This is h t can in equilibrium be written as n equivalent to saying that (2) must hold at all time instants. ` mg mv h = 0 (4) n n `=2 2 which gives us the no-rollover condition ` T (v ; 0; ) : (5) n n 2h G Tab. I shows that the kinetic tire friction coefficient is h typically slightly lower than the static stability factor`=2h (the average static stability factor among vehicle models introduced mv n n in 2003 was approximately 1:4 for passenger cars and 1:2 for mg SUVs 28), and hence, the no-sliding condition should be violated before the no-rollover condition. That being said, a more realistic model including e.g., normal forces or friction Fig. 3. To avoid a rollover event, the torquemg`=2 from the gravity force forces unevenly distributed among the contact points between must exceed the torquemv h from the centripetal force. This is equivalent n n to saying that (5) must hold at all time instants. the tires and the road surface 29, 30, nonzero pitch or roll angles 26, or tripped rollovers 31, would complicate the analysis. ? i.e., F = mg, where m denotes the mass of the car, and g Now, we will identify cornering events by observing is the gravitational acceleration at the surface of the earth. whether either of the statistics in (2) and (5) reach some Assuming that the car at time t travels with speed v and n n predetermined thresholds, that is, whether the driver violates longitudinal acceleration a , in a perfect circle with radius n the condition v = , the horizontal force F on the tires, can be divided n n h into two parts (see Fig. 2). The first is the centripetal force, C : T (v;a;) and T (v; 0;) (6) SL RO equal tomv and directed towards the center of the circle. n n where = c  and = c `=2h for some constants SL 1 RO 2 The second is the tangential forcema , directed along the line n c ;c 2 (0;1). These constants will have to be tuned to suit 1 2 tangent to the circle at the position of the vehicle at time t . n the needs of the particular application at hand, and to comply Since these two forces are perpendicular, and since we must with the desired level of risk that should result in a cornering haveF =F if we are to avoid slipping, the friction equation f h event. As motivated above, the study will for simplicity focus can be rewritten as  on the estimation of T = T (v ;a ; ) (i.e., we will set n n n n c = 1). Hence,C is tested by first estimating the input T (v ;a ; ) (2) n n n 2 variables as v , a , and , and then calculating the njn njn njn ?  b where the force ratio F =F is denoted by h corresponding test statistic T =T (v ;a ; ). n njn njn njn When there is no currently ongoing estimated cornering p 1  ? 2 2 2 T (v;a;) =F =F = v +a : (3) h event for some given detection threshold , the start of an SL g b event is defined to be the first time t for which T  . n n SL s s Correspondingly, the condition for the avoidance of a Similarly, the end of an event starting at t is defined as the n s b rollover can be found by studying the moment equation around time t , where n is the largest integer such that T  , n e n SL e e b G, the car’s center of gravity (see Fig. 3). At the moment of and where T  0:35 for each n such that n  n n . n s e a rollover, the innermost wheels will have left the ground, By requiring the test statistic to fall below 0:35 (which is and the normal force only operates on the outer wheels. The considered to be a typical value of T while cornering in a force has an associated lever arm of `=2 (where ` is the track non-aggressive manner) in between two separate detections, width of the car), giving a torque of mg`=2. Including also we are able to avoid the risk of obtaining an increasing number the contribution from the radial force and denoting the height of detections as is increased. Without this adjustment, the SL above ground of G by h, the total moment around G at time reference system (described in Section V-A) will often divide,4 what intuitively should be classified as one event, into multiple p in the state vector, we avoid modeling  , and will also n n events, as the test statistic fluctuates around . We will follow limit the width of the temporal correlation of the measurement SL the convention of denoting the duration of a cornering event errors to one sample period. Using that w is a moving p n detected using reference data and data from a smartphone by average process of first order, we augment the state vector ( ; ) and (t ;t ), respectively. with the measurement errors w and w , and discretize m m n n p p 1 s e s e n n Given that a cornering event has been detected, the associ- (8) as (see 33) ated estimated risk level will be defined as x =f(x ) +Gq (13) n+1 n n  max b b T = maxfT : t t t g (7) n n n n t :t s e n n s e where 2 3 2 3 R t N (1) n+1 whereft g are the sampling instances of the smartphone. v(t) cos((t))dt n p n=1 n t n max 6 7 6 R 7 The true risk level, denoted byT , will be approximated t (2) n+1  : m m 6 7 6 7 s e p v(t) sin((t))dt n t 6 7 6 n 7 by the risk level resulting from applying the analogous defi- 6 7 6 7 v t v n n 6 7 6v + (e 1)= a 7 n v n nition to the test statistic provided by the reference system. 6 7 6 7 a t n v n 6 7 6 7 e a n 6 7 6 7  t n  n 6 7 6 7 III. FILTERING OF GNSS MEASUREMENTS  + (e 1)=   n  n 6 7 6 7 x = ; f(x ) = ; n n n t 6 7 6  n 7 e n This section will describe the method of estimating the 6 7 6 7 (1) 6 w 7 6 7 p n b 0 input variables required for the calculation of T . Standard 6 7 6 7 n (2) 6 7 6 7 2 w deviations are denoted by , where the associated stochastic p 0 6 n 7 6 7 () 6 7 6 7 (1) (1) variable is identified by the subscript. The identity and zero 6 7 6 7 w p w n1 p n 4 5 4 5 matrix of size kk are denoted by I and 0 , respectively, k k (2) (2) w p w n1 p n while 0 denotes the zero matrix of size k k . k ;k 1 2 1 2 (14)  A. State-space Model and t = t t . In the implementation, the integrals n n+1 n in the topmost rows have to be approximated in terms of x , Denoting the vehicle’s position and bearing by p(t) = n   (1) (2) with the process noise q adjusted accordingly. The applied p (t) p (t) and (t), respectively, the vehicle dynam- n procedure is presented in Appendix I together with the process ics are modeled by  noise covariance matrixQ = Cov(q ) and the process noise n n x _ (t) =f (x (t)) +q(t) (8) gain matrix G. c c c The measurement equation is given by where 2 3 2 3 v(t) cos((t)) y =Hx +w (15) n n n p(t) 6 7 v(t) sin((t)) 6 7 6 7 v(t) h i   6 7 6 7 a(t)     (1) (2) g 6 7 6 7 where y = ; H = H H , x (t) = a(t) ; andf (x (t)) = ; (9) p v  n n n c c c n 6 7 6 7 a(t) v 4 5 6 7 (t)     4 5 (t)  I 0 0 0  I I 3 3;1 3;1 3;1 2 2 (1) (2) (t) H = ; and H = : (t)(t)  0 0 1 0 0 0 1;3 2 2 with the decay factors ; 2 (1; 0). Further, q(t) v  We will now study the measurement error covariance matrix. is assumed to be Gaussian white noise with covariance For ease of notation, we collect the doppler-based measure-    f Cov(q(t);q()) =(t)Q, where ments of speed and bearing in the vector do = v  . n n n   The exact process of calculating these measurements is typi-  2 2 Q = diag 0 0 0  0  ; (10) q q v  cally unknown. However, given the methods presented in the f literature 34, 35, we will assume that do is derived from () denotes the Dirac delta function, and diag() denotes the n a preceding measurement of the two-dimensional velocity matrix which has diagonal elements equal to the argument.   (1) (2) The GNSS position measurements p are typically subject v = v v . The measurements are modeled according n n n n to temporally correlated errors, and will be modeled by to v = v +w , where v denotes the true velocity, and n n v n n 2 w is assumed to be white noise with covariance  I . v 2 n w v p =p + +w : (11) n n n p n Disregarding the necessary extensions of the arctan-function f required to make (t) continuous, do can be approximated n The error term  represent a slowly varying bias, and we n by will use that  . The remaining errorw is assumed n n1 p n 2 to be white noise with covariance  I . Since we are not 2 w f p do =f (v ) n do n interested in estimating any absolute position of the vehicle, (16) f (v ) do n it is convenient to work with the relative position f (v ) + (v v ) do n n n v n  p =p p : (12) p  n n n1  (1) (2) 2 2 (1) (2) wheref (v ) = is as- (v ) + (v ) arctan(v =v ) do n n n n n g Note that p  p +w where the measurement error n p sumed to be unbiased, andf (v )=v denotes the Jacobian n n do n n  w =w w . Hence, by including p instead of of f . We then further have p p p n do n n n15   f (v ) f (v ) Algorithm 1 : Estimation of Vehicle Dynamics do n do n 2 f Cov(do ) n w v v v n n 1: Initialize x and P . 1j1 1j1   (17) 1 0 2 2: for n = 1 :N do = : w 2 v (i) 0 1=v 40 n 3: Generatef g according to (20). i=0 njn (i) (i) (i) Using this, we can approximate the covariance matrix of the 4: x =f(x ) +q n+1jn njn njn P 40 (i) measurement noise by 5: x = w x i n+1jn i=0 n+1jn    (i) (i) 2 2 2 6: y =Hx R = Cov(w ) = diag 0 0   =v : (18) n n w w n v v n+1jn n+1jn P 40 (i) 7: y = w y i n+1jn i=0 n+1jn Noteworthy, even though the vehicle’s approximate position (y) 8: P =hy ;y i +R n+1jn n+1jn n is needed when estimating the velocity from doppler measure- n+1jn (xy) ments, the resulting error dependence can be neglected since a 9: P =hx ;y i n+1jn n+1jn n+1jn (xy) (y) position error of about 100 m only will give a velocity error 1 10: K =P (P ) n n+1jn n+1jn in the order of 0:01 m=s 34. (i) (i) (i) 11: x =x +K (y y ) n n+1 n+1jn+1 n+1jn n+1jn  P 40 (i) 12: x = w proj x n+1jn+1 i i=0 n+1jn+1 B. Unscented Kalman Filter (x) 13: P =hx ;x i n+1jn+1 n+1jn+1 n+1jn+1  The measurements are filtered in an unscented Kalman b 14: T =T (x ) n+1 n+1jn+1 filter with reformulated correction steps, the process noise 15: end for augmented into the state vector, and where the state estimates All steps in Algorithm 1 refering to a sigma point i are looped over all are constrained using projections 36. The augmented state sigma points. vector is denoted by   projected onto the spacefx:T (x) g according to  x proj n  = (19) n q n (i)  (i) 2 proj(x ) = arg minkx x k 1 proj n+1jn+1 n+1jn+1 P njn x and the propagated sigma points are given by proj (23) 8 (0) s:t: T (x ) proj proj  = njn njn  (i) () 2 1=2  = +(P ) ; i2f1;:::; 20g (20) where kxk = x Mx. Note that this ensures that :;i M njn njn njn T (x )  since T is convex. The projection can : n+1jn+1 proj (i+20) () 1=2  = (P ) ; i2f1;:::; 20g :;i njn njn njn be found by numerical means using gradient-based methods. The choice of only applying the constraint to updated sigma where " points has previously been motivated in 39 and references (x)     P 0 () 10 njn therein. x 0  = 1;10 ; and P = : (21) njn njn njn 0 Q 10 n The parameter  describes the trade-off between, on proj the one hand, minimizing the risk of incorrectly altering the (c) Using measurements up until t , c and P denote n njn estimates, and on the other hand, utilizing the implicit outlier 2 1 2 njn 1 2 the estimate ofc and the associated error covariance matrix, n detector resulting from the use of (23). In the implementation, 1 1=2 respectively. Further, (M) and M denote the Cholesky :;i we used  = 0:9 which is a rather conservative choice proj decomposition and the i:th column of the matrix M, respec- given that the detection threshold typically will be chosen far tively. The filter algorithm is presented in Algorithm 1 where below this value. In light of the limitations implied by Tab. the inner product is defined by I, a more refined implementation, applied to driving under 40 more varied sets of circumstances than what is described in X  (i) (i) Section V, would in all probability benefit from letting  hc ;d i = w (c c )(d d ) ; proj i njn njn njn njn 1 2 1 2 njn 1 2 njn 1 2 1 2 1 2 be dependent on the vehicle’s speed or the weather. i=0 (22)  and we use that T (x) = T (v;a;) where v;a; and are C. Filter Tuning elements in x. The weights were set as w = 1=41 for i2 i p Assuming that the accuracy of the doppler measurements f1;:::; 41g, which further gives = 41=2. Refer to 33 and is fairly similar among different smartphone models, we will 37 for details on unscented Kalman filtering. 2 2 2 use  = (0:2 m=s) 40. Likewise, we set  = w w v p G-forces in the vicinity of 1g are almost exclusively at- 2 (1:5 m=s) . (Note that this only corresponds to the propor- tributed to sports cars with high performance tires, and most tion of the position errors which are temporally independent.) production cars never reach these levels even during aggressive The decay factors are typically hard to estimate from data, cornering. (For comparison, see the skid pad numbers, i.e., and will be fixed at =0:5 1=s and =0:1 1=s. v  the highest obtained g-forces during driving along a circle of Since cornering events are characterized by relatively dynamic 100 m, collected in 38 from tests in the magazine ”Car and driving, this should be reflected in the model parameters. (i) Driver”.) Therefore, all updated sigma points x are n+1jn+1 While a small value on correspond to aggressive driving, v6  5 s  + 5 s t t n n m m s e s e is now defined as FALSE t h i ALARM   N N b R() =E LfT (;  ;  )g ;fT ( )g (24) n 1 2 n 1 n=1 n=1 Z MISSED ESTIMATED EVENTS  DETECTION N N b = LfT g ;fT g p ( ; )d d n n  ; 1 1 1 2 n=1 n=1 1 2   m s m e t whereL is a loss function adjusted to the insurers preferences. The pdf p ( ; ) will in practice have to be estimated  ; 1 2 1 2 TRUE EVENTS from empirical data, and will inevitably be subject to design choices related to driving behavior, road type, the chosen Fig. 4. An estimated event with duration (t ;t ) can only be mapped to n n s e a true event with duration ( ; ) if t 2 ( 5s; +5s) smartphone, etc. Note that we have deliberately separated the m m n m m s e s e for some t 2 (t ;t ). The mapping between estimated and true events n n n s s parameters in , describing the driving behavior, from  , to 1 is performed so as to bijectively map the largest possible number of events. emphasize that the true vehicle dynamics are not generated from (8). is limited from below by the bandwidth of the modeled v The performance evaluation in this article will be limited car dynamics (which typically is less than 2 Hz 24). to studies of the estimator presented in Section III, in terms The process noise parameters  and  are considered q q v  ofR( ; ). The Bayesian risk will be estimated by con- q q v  to be design parameters and their influence on the estimator sidering the available reference data to be representative of is studied in the subsequent section. The appropriate value set  p ( ; ), i.e., by using thatR()L(), where  ; 1 2 1 2 on the parameter  , describing the smoothness of the ride q v in terms of longitudinal acceleration, will be dependent on the M X   (j)  N  N 2  b driver, the intended application, etc. Similarly,  will to a L() = 1=M LfT (;  ;  g ;fT ( )g ); n n q 1 2 n=1 1 n=1  large extent depend on the road type and the chosen speed. We j=1 p 2 (25) will study values in the intervals 0  0:8 m=s  Hz q v p  and 0  1 1=s Hz. q  M denotes the number of smartphones,  representes the 1 (j) outcome of  provided by the reference data, and  de- 1 2 IV. FRAMEWORK FOR PERFORMANCE EVALUATION notes the sensor errors in the measurements from smartphone  We will now describe the general methodology used to j.T ( ) will be approximated by the corresponding estimate n 1 b evaluate the performance of the estimatorT . This will provide provided by the reference system. n We will study three different loss functions, the first being a common ground for the performance evaluation in Section defined as V, and illustrate how the study can be extended based on the L =dMD + (1d)FA (26) intended application. The presented framework can be used to MDFA compare the performance of any given estimators ofT , or to n with d2 0; 1, and where MD and FA are the number of calibrate a parameterized estimator. missed detections and false alarms, respectively, divided by the We first specify how to map the detections provided by total number of events. Moreover, assuming that the detected a smartphone with those provided by the reference system event with duration (t ;t ) has been mapped to the true n n s e (considered to be the true cornering events). Definitions of event with duration ( ; ), we introduce the loss functions m m s e a missed detection and a false alarm will then follow in a q   straightforward manner. The objective of the mapping is to link max max 2 b L = E (T T ) (27) RMSE t :t  : n n m m s e s e each estimated event with the underlying physical event, if any, that it has detected. After calculating the start and end points and   max max b of each cornering event, a subset of the events provided by L =E T T (28) BIAS t :t  : n n m m s e s e a smartphone is bijectively mapped onto the events provided where the expectation is taken over all true events detected by the reference system. The mapping is constructed so as by the reference system. If a smartphone did not detect a to map the largest possible subset of events, subject to the max b specific event with duration ( ; ), T is defined as m m s e t :t n n s e constraint that each event with duration (t ;t ) only can n n s e b the maximumT such thatt 2 ( 5 s; + 5 s) and n n m m s e be mapped to an event with duration ( ; ) if t 2 m m n s e wheret 62 (t ;t ) for any detection of the smartphone with n n n s e ( 5 s; + 5 s) for somet 2 (t ;t ) (see Fig. 4). m m n n n s e s e duration (t ;t ). n n s e The events detected by a smartphone and the reference system which has not been mapped are considered to be false alarms V. FIELD STUDY and missed detections, respectively. A. Experimental Specification Now, consider a generic driving trip with data collected at N the sampling instancesft g . The vehicle dynamics and The estimation framework presented in Section III was n n=1 the sensor errors (together with sampling instances) of the applied to 31 minutes of data collected during aggressive trip are described by the stochastic variables  and  , driving under normal road conditions (see Tab. II). Reference 1 2 respectively. Their joint probability density function (pdf) is data was collected using a Microstrain 3DM-GX3-35, which denoted byp ( ; ). The Bayesian riskR() (where  includes an IMU and a GNSS sensor with update rates of  ; 1 2 1 2 denotes the parameter set describing the parameterization of 100 Hz and 4 Hz, respectively. The IMU and GNSS data b b the estimatorT ) associated with the estimatorT (;  ;  ) were then fused in a GNSS-aided INS, from which the force n n 1 27 TABLE II TABLE III SUMMARY OF THE DATA USED IN THE FIELD STUDY. MISSED DETECTIONS AND FALSE ALARMS Elapsed time min 31 S3 X2 S4 Average Number SL of events Elapsed distance km 17 max Number of events with T 0:35 40 Missed detections %  : m m s e max (L with d = 1) Number of events with T 0:5 29 MDFA  : m m s e max Number of events with T 0:65 21 0:5 7 31 0 13 29  : m m s e max Number of events with T 0:75 7 0:55 17 48 14 26 28  : m m s e 0:6 24 44 12 27 26 0:65 37 47 26 37 21 Time length of harsh cornering 0:7 62 38 31 44 12 0.6 0:75 67 17 33 39 7 0.4 False alarms % 0.2 (L with d = 0) MDFA ¯ 0 0:5 24 7 34 22 29 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0:55 14 3 24 14 28 γ SL 0:6 12 4 24 13 26 Fig. 5. The average length of time during which the test statistic exceeds . SL 0:65 21 11 47 26 21 0:7 38 23 69 44 12 Receiver operating characteristics 0:75 100 83 150 111 7 1 (Missed detections + False alarms)=2 % (L with d = 1=2) 0.8 MDFA γ SL 0:5 16 19 17 17 29 0.55 0:55 16 26 19 20 28 0.6 0:6 18 24 18 20 26 0.6 0.65 0:65 29 29 37 32 21 0.7 0:7 50 31 50 44 12 0.75 0:75 83 50 92 75 7 0.4 ¯ Missed detections and false alarms as a percentage of the p 2 number of events, with  =0:4m=s  Hz and q v p  =0:41=s Hz. q  0.2  two calculations of time length). It can be seen that t( ) SL decreases in a linear fashion as 0:5  0:7. Due to the SL 0 0 0.2 0.4 0.6 0.8 1 comparatively low update rate of the smartphones (1 Hz), the ¯ L with d = 0 MDFA resulting test statistic will often fail to capture short events of harsh cornering, and the risk level of detected events will tend Fig. 6. Receiver operating characteristics of the detector with  = q v p p to be underestimated. The estimated risk level can be increased 2 0:4m=s  Hz and  =0:41=s Hz. q  by the use of a larger noise variance, however, this is done at the expense of resistance to false alarms. ? ratio F =F was calculated (estimated from accelerometer h Since deliberate attempts were made to reach large g-forces, measurements after subtraction of the gravitational accelera- the exact average time lengths displayed in Fig. 5 should not tion and the estimated sensor bias). To minimize the influence be interpreted as typical of normal driving. of high frequency errors due to sensor noise, inadequate separation of the gravitational and spatial acceleration, etc., B. Performance Evaluation the resulting estimate of T was low-pass filtered with a cut- off frequency of 1=2 Hz. Data was simultaneously collected Since there is no known industry standard for the detec- from GNSS sensors in the three Android phones Samsung tion threshold , we will study thresholds in the interval SL Galaxy S3, Samsung Galaxy Xcover 2 (abbreviated X2), and 0:5  0:75. According to the authors’ experience, this SL Samsung Galaxy S4. corresponds to values which are typically obtained during what  Fig. 5 shows the average length of timet( ) during which intuitively should be classified as aggressive driving. SL the test statistic T , calculated using the reference system, An indication of the performance that can be expected n stays above the detection threshold after exceeding it is given in Tab. III, which shows the obtained missed de- SL (note that this is not the average time length of a cornering tections and false alarms for different thresholds, with the p 2 event since T is not required to fall below 0:35 in between design parameters fixed at  = 0:4 m=s  Hz and n q v 1−L with d = 1 MDFA t(γ )s SL8 (a) Missed detections (a) Missed detections 1 1 −3/2 −5/2 σ s σ ms q q θ v 0.8 0.8 0.1 0.1 0.4 0.4 0.6 0.6 0.8 0.8 0.4 0.4 0.2 0.2 ¯ ¯ 0 0 0.5 0.55 0.6 0.65 0.7 0.75 0.2 0.4 0.6 0.8 1 √ γ SL σ 1/s· Hz q θ .8 .8 (b) False alarms (b) False alarms 2 0.5 −3/2 −5/2 σ s q σ ms q θ v 0.4 1.5 0.1 0.1 0.4 0.4 0.3 0.8 0.8 1 0.2 0.5 0.1 ¯ ¯ 0 0 0.5 0.55 0.6 0.65 0.7 0.75 0.2 0.4 0.6 0.8 1 √ γ SL σ 1/s· Hz q θ .8 .8 (c) (Missed detections and False alarms)/2 (c) (Missed detections + False alarms)/2 1.2 0.5 −3/2 −5/2 σ s q σ ms θ q 1 v 0.1 0.4 0.1 0.8 0.4 0.4 0.8 0.6 0.8 0.3 0.4 0.2 0.2 ¯ ¯ 0 0.1 0.5 0.55 0.6 0.65 0.7 0.75 0.2 0.4 0.6 0.8 1 √ γ SL σ 1/s· Hz q θ Fig. 7. Missed detections (a), false alarms (b), and the sum of missed Fig. 8. Missed detections (a), false alarms (b), and the sum of missed detections and false alarms (c), divided by the total number of true cornering detections and false alarms (c), divided by the total number of true cornering events. The process noise variance of the vehicle speed was set to  = q v events. The detection threshold was set to =0:6. p SL 2 0:4m=s  Hz. p the corresponding false alarms, this is done at the expense of  = 0:4 1=s Hz. For detection thresholds in the range q  a larger number of false alarms. The collected data indicate of 0:5  0:6, the sum of missed detections and false SL that the total sum of missed detections and false alarms stays alarms is around 40% of the total number of cornering events. approximately the same at smaller thresholds (see Fig. 7 (c)). For higher thresholds, the performance deteriorates. In Fig. 8 (a), we study the missed detections as depending The performance of the estimator, as depending on the on both  and  , with the detection threshold fixed at detection threshold, is further illustrated by the receiver op- q q v  0:6. Note that the number of missed detections approaches erating characteristics (ROC) displayed in Fig. 6. The ROC 100 % as approaches 0, that is, when the filter has a large with = 0:5 is identical to the ROC with = 0:55 (the SL SL q  resistance to sudden steering maneuvers. Similarly, Fig. 8 (b) latter threshold only results in one less true cornering event shows that the number of false alarms decreases to zero as than the former). Once again, the performance is seen to be q  approaches 0. Disregarding the above mentioned examples, the comparable for thresholds in the range of 0:5  0:6, SL performance is robust with respect to the design parameters, but then declines as the threshold is increased. Note that, as p 2 and the two plots for which  = 0:4 m=s  Hz and opposed to all other studies in this section which employs the q v p 2  = 0:8 m=s  Hz display no strong dependence on same threshold for estimated and true events, each curve in q v neither of the parameters (see also the sum of missed detec- Fig. 6 is obtained by fixing the threshold for the true events tions and false alarms in Fig. 8 (c)). Since there is a large at , while varying the threshold for the estimated events. SL Now, consider Fig. 7 (a), where the missed detections, range of values of and with comparable performance, q q v  averaged over the three smartphones, are displayed as a parameter calibration is expected to be straightforward in function of the threshold. It can be seen that the number practical implementations. of missed detections to some extent can be decreased by Fig. 9 (a) shows that the root mean square error of the increasing the value of  . (The parameter  determines estimated risk level is around 0:12, and hence, the normalized q q   the well-known trade-off between dynamic response and noise root mean square error is slightly above 10%. Moreover, the resistance associated with the measurements of bearing, and bias of the estimated risk level is essentially zero (see Fig. hence, to a large extent also controls the estimated angular 9 (b)), disregarding the previously mentioned case when  q  velocity .) However, as seen in Fig. 7 (b) which shows approaches zero. njn L with d = 1/2 L with d = 0 L with d = 1 MDFA MDFA MDFA L with d = 1/2 L with d = 0 L with d = 1 MDFA MDFA MDFA9 APPENDIX I (a) Root mean square error of estimated risk level 0.18 −5/2 σ ms We denote the covariance and gain matrix of the process q v 0.16 0.1 noise (see (21)) by 0.4 "   0.8 (1) 0.14 Q 0 I   n 2 8 ¯ Q = ; and G = ; (29) n (2) 0 0.12 0 Q 2;8 2 n respectively. The submatrices are defined by 0.1 0.2 0.4 0.6 0.8 1 √ 2 3 (p) (p;v) (p;) σ 1/s· Hz q θ Q Q Q n n n  6 7  (1) (p;v) (v) (2) 2 Q = ; and Q = I : 4 Q Q 0 5 2 n n n 2 n w (b) Bias of estimated risk level p  (p;) () 0.05 Q 0 Q n 2 n (30) 0 First, note that 41 −5/2 −0.05   σ ms q 2 v t 2 t 2  j n j n (e 2) + 2 t 1 (e 1)  q j j n j 0.1 (j) Q = n t 2 2 2 t 3 j n j n −0.1 0.4 (e 1) (e 1) 2 j j j 0.8 (31) −0.15 0.2 0.4 0.6 0.8 1 for j = v; . Now, integrating the true integral equal to √ σ 1/s· Hz q θ the relative position p , linearizing the integrand, and n+1 approximating the integral according to the trapezoidal rule, Fig. 9. Average root mean square error (a) and average bias (b) of the estimated risk level. The detection threshold was set to =0:6. SL we obtain   Z t n+1 cos((t)) p = v(t) dt n+1 sin((t)) t n VI. CONCLUDING REMARKS Z   Z   t t n+1 n+1 cos( ) cos( ) n n  v dt + (v(t)v ) dt n n sin( ) sin( ) n n t t n n This paper has presented a framework for the detec-   Z t n+1 sin( ) tion of dangerous vehicle cornering events, well-suited for n + ((t) )v dt (32) n n cos( ) smartphone-based insurance telematics applications. The abil- n t n      ity to detect cornering events exceeds the capacity of current t n cos( ) sin( ) n n  v v  n n n telematics solutions based on on-board-diagnostics, and en- sin( ) cos( ) 2 n n    ables both new ways to differentiate among drivers and new cos( ) v sin( ) v n n n n+1 value added services. The detection is based on a continuously : sin( ) v cos( )  n n n n+1 updated test statistic estimated using an unscented Kalman   filter applied to GNSS measurements of position, speed, and v  Rewriting according to (13) we arrive at n+1 n+1 bearing.   q n A general framework for performance evaluation and es- 3 p =f (x ) +S (33) n+1 p n q n timator calibration was presented as depending on a generic 5 loss function. Three loss functions were introduced: the first where f (x ) replaces the topmost rows of f(x ) in the p n n designed for an application aiming to minimize the number implementation, c denotes element n in the vector c, and n of missed detection and false alarms of cornering events; and   the other two designed for the estimation of the risk level of t cos( ) v sin( )  n n n n S = : (34) a cornering event. sin( ) v cos( ) 2 n n n The performance of the estimator was evaluated in a field The corresponding process noise is given by study where data was collected from three smartphones and " a reference system utilizing high rate sensors. It was shown (v)  Q 0 n (p) 1;1 that the expected number of missed detections and false alarms Q =S S (35) n () 0 Q n 1;1 sum up to around 40% of the total number of cornering events, while the estimated risk level has a normalized root mean with M denoting the element at rown and columnn n ;n 1 2 1 2 square error of approximately 10%. 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