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Vehicle tracking using a microwave radar for situation awareness
ARTICLE IN PRESS Control Engineering Practice 18 (2010) 383–395
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Control Engineering Practice journal homepage: www.elsevier.com/locate/conengprac
Vehicle tracking using a microwave radar for situation awareness Seongkeun Park a, Jae Pil Hwang a, Euntai Kim a,, Hyung-Jin Kang b a b
School of Electrical and Electronic Engineering, Yonsei University, C613, Sinchon-dong, Seodaemun-gu, Seoul 120-749, Korea Mando Central Research Center, Gyeonggi-do 449-901, Korea
a r t i c l e in fo
abstract
Article history: Received 27 December 2008 Accepted 7 December 2009 Available online 18 January 2010
In this paper, a probabilistic target vehicle tracking method is proposed for situation awareness of intelligent cruise control (ICC) vehicle. The ICC vehicle considered herein is equipped with a 24 GHz microwave radar for tracking the preceding vehicle. To overcome the severe dispersion and noise of the microwave radar, a statistical model for the radar is built and it is applied to the hybrid particle filter. The hybrid particle filter is combined with the interacting multiple models (IMM) to track the preceding vehicle and predict the driver’s intention. Furthermore, the modified hybrid particle filter is proposed to cope with the missing or multiple measurements of the microwave radar. Finally, a computer simulation is conducted and the validity of the proposed method is demonstrated. & 2009 Elsevier Ltd. All rights reserved.
Keywords: Microwave radar Scattering IMM Hybrid particle filter EM
1. Introduction The majority of automobile companies have recently shifted their research focus from the classical passive safety system such as safety belt (Chan, 2007; Watanabe, Umezawa, and Abe, 1994) to various active driver assistance systems. In particular, intelligent cruise control (ICC) including adaptive cruise control (ACC) and stop-and-go has received considerable attention within the vehicular community with the recent advances of monolithic microwave integrated circuit (MMIC), integrated circuit (IC) (Naranjo, Labayrade, Royere, Hautiere, and Aubert, 2007; Russell et al., 1997; Tsang, Hall, Hoare, and Clarke, 2006) and computers (Alessandretti, Broggi, and Cerri, 2007;Forsyth and Ponce, 2002; ¨ Holzmann, Halfmann, Germann, Wurtenberger, and Isermann, 1997;Hwang, Rou, Park, Kim, and Kang, 2006;Moon, Moon, and Yi, 2009;Perrollaz et al., 2006). Fig. 1 shows the configuration of an ICC system (Kim and Hong, 2004). The performance of an ICC system depends on how accurate the sensors are in recognizing traffic and obstacles ahead. The sensor candidates include a lidar (Guang and Tomizuka, 2003; Takagi, Morikawa, Ogawa, and Saburi, 2006), a vision (Sotelo et al., 2004; Stein, Mano, and Shashua 2003) and a radar (Eriksson and As, 1997; Moon and Yi, 2002). When compared with the lidar, the microwave radar has the following properties: (1) Microwave radar is more robust in all weather conditions including fog and rain. (2) Microwave radar can measure the relative velocities of the nearby objects using Dopplers’ effect (Russell et al., 1997) and Corresponding author.
E-mail address: [email protected] (E. Kim). 0967-0661/$ - see front matter & 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.conengprac.2009.12.006
therefore is widely used as a range sensor in most of the commercial ICC systems. (3) In general, the price of radar used to be much higher than that of lidar. But with the rapid growth of the silicon semiconductor technology, the radar can be manufactured not only using chemical semiconductor such as GaAs (Larson, Hackett, and Lohr, 1991) but also using silicon semiconductor such as SiGe (Bock et al., 2004). Thus, the price competitiveness of the radar gets improved against that of the lidar these days. Thus, today the price of radar is only slightly more expensive than that of lidar. (4) The radar can be installed behind a vehicle bumper or other cover, but the lidar must be installed in an exposed area. Thus, radar is preferred to lidar from the perspective of design. (5) The relative velocities of preceding vehicles can be measured directly by radar. Lidar, however, cannot measure relative velocity, which therefore must be computed by differentiating relative distances. When compared with the vision, the microwave radar has the following properties: (1) In general, the vision cameras are cheaper than microwave radar and are still capable of returning rich environmental information. (2) However, the vision is much less robust in the context of light and weather variations, and range measurement is inaccurate. (3) Information from the vision cameras is so rich that tremendous computational burdens are required to localize vehicles within images. (4) Consequently, the vision cameras are recommended only as secondary sensors to millimeter radars.
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Fig. 1. The configuration of an ICC system.
In summary, microwave radar is a key sensor in the ICC at present and will continue to remain so in the future. Some previous works have reported the studies about vehicle tracking using a microwave radar (Gresham et al., 2004; Kim and Hong, 2004; Stursberg, Fehnker, Han, and Krogh, 2004; Venhovens, Naab, and Adiprasito, 2000). To my knowledge, however, most of the previous works treated the radar as a sensor with Gaussian noise, and none of them fully considered the dispersive and noisy nature of the microwave radar when it is used for target tracking. In this paper, a target vehicle tracking method is presented for the ICC using a microwave radar. To deal with the complicated characteristics of the microwave radar, a statistical model of the radar is built and it is combined with the hybrid particle filter to track the target vehicle. This paper is a companion paper of Park, Hwang, Kim, Lee, and Jung work (2010), in which the same radar was used for target identification. The remainder of this paper is organized as follows: In Section 2, microwave radar used herein is introduced and the problem is described in detail along with the solution requirements In Section 3, the radar measurements are statistically modeled. In Section 4, a modified hybrid particle filter is proposed to estimate the state of the preceding vehicle. In Section 5, simulation is conducted. Finally, some conclusions are drawn in Section 6.
Fig. 2. The ICC vehicle equipped with a 24 GHz microwave radar.
Fig. 3. A driving situation.
2. Problem formulation 2.1. Microwave radar and adaptive cruise control The microwave radar used in this paper, MASRAU0025, is manufactured by M/A-COM and is composed of a 24 GHz radar sensor and microprocessor unit (Manual of MA-COM MASRAU0025). The MASRAU0025 detects the objects up to 30 m ahead and returns a variety of information about the object including the distance, azimuth angle, velocity, etc. The ICC vehicle is equipped with the microwave radar MASRAU0025, as shown in Fig. 2. As shown in figure, MASRAU0025 is mounted on the car and it is connected to a laptop computer through a control area network (CAN). The laptop computer has a P-IV processor and the program is operating on Windows XP. Fig. 3 shows the driving situation considered in this paper in which a target vehicle and the ICC vehicle are traveling on a three-lane road. The target vehicle changes the lane in a potentially dangerous manner and passes the ICC. The ICC should track the preceding target vehicle and identify the intention of the driver via the microwave radar. 2.2. Problem description and solution Here, the vehicle tracking problem will now be described in detail.
Fig. 4. Points of a vehicle detected by a 24 GHz radar.
(1) The microwave radar easily disperses and the detection points on the vehicle that the radar detects will vary every time. Mainly, the radar detects the left corner or right corner of the vehicle ahead, but occasionally, the radar detects points in between, as shown in Fig. 4. When the radar detects the
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in-between points, the measured point is almost random between the two corner points. (2) For a single scan, the number of measurements that the microwave radar may return for a target vehicle varies from zero to three. With the chance of over 80%, the radar returns a single measurement. Occasionally, however, the radar returns two or three measurements for a target or does not return at all. (3) The intention of the driver should be predicted from the movement of the preceding target vehicle, for example, whether the vehicle is moving uniformly or changing lanes. This vehicle tracking problem does not allow for typical solutions due to the above difficulties. More specifically, the Kalman filter, or its variants, cannot be applied due to the difficulties described in the radar and combine it with the particle filter (PF). To meet requirement (3), the PF is combined with the interacting multiple model (IMM) to predict the driver’s intention. The result is a hybrid particle filter with both continuous and discrete states.
In this section, a statistical model is build for the microwave radar MASRAU0025. The statistical model is utilized in tracking a preceding target vehicle in a probabilistic way. As stated, MASRAU0025 detects the left corner or right corner of the vehicle, or the middle part in between, mostly more than once. and Z ¼ fzi gN First, two sets of data X pos ¼ fxpos;i gN i ¼ 1 are i¼1 collected using the MASRAU0025, where xpos;i denotes the true position of a car and zi denotes the associated measurement that microwave radar returns. Then, the measurement error is defined as ð1Þ
and its statistical model is built from the data set W ¼ fwi gN i ¼ 1. The measurements from the radar are modeled using a mixture of three components and the parameters of the model are identified by the Expectation and Maximization (EM) (Bishop, 2006). The mixture is composed of two Gaussians for the left and right edges and a uniform for the center face in between. The measurement model is X pj pj ðwjHj Þ pðwjHÞ ¼ j ¼ fleft;center;rightg
¼ pleft Nðwjlleft ; Rleft Þ þ pcenter UðwjAcenter Þ þ pright Nðwjlright ; Rright Þ
Table 1 The algorithm of building vehicle measurement model. Algorithm building measurement model ðWÞ 1. Initialize H ¼ fp; Hleft ; Hcenter ; Hright g 2. For all wi , assign the data to each component by
pj pj ðwi Þ
tij ¼ P
k ¼ fleft;center;rightg
pk pk ðwi Þ
3. Update the parameters by
lleft ¼
N 1 X t ðw Þ Nleft i ¼ 1 i;left i
Rleft ¼
N 1 X t ðw lleft Þðwi lleft ÞT Nleft i ¼ 1 i;left i
lright ¼
N 1 X t ðw Þ Nright i ¼ 1 i;right i
Rright ¼
N 1 X t ðw lright Þðwi lright ÞT Nright i ¼ 1 i;right i
Acenter *fwi jti;center Z eg
3. Measurement model of microwave radar
wi ¼ zi xpos;i
385
ð2Þ
where pleft ðÞ and rightðÞ are the Gaussian distributions NðÞ and pcenter ðÞ is the uniform distribution UðÞ. pleft ¼ pðj ¼ leftÞ is a prior probability that this detection is from the left edge and pcenter and pright are defined in the similar way. Further, P j ¼ fleft;center;rightg pj ¼ 1. pleft ð dÞ is a distribution of w when the microwave radar detects the left edge of the preceding car and centerðÞ and pright ðÞ are defined in the similar way. Further H ¼ fp; Hleft ; A Hcenter ; Hright g, p ¼ fpleft ; pcenter ; pright g, Hcenter ¼ ðAcenter Þ, Hright ¼ ðlright ; Rright Þ; lj and Rj are the mean and covariance of the Gaussians, respectively, and Acenter is the area on which measurements are detected uniformly. The parameters of the measurement model can be estimated by the Expectation–Maximization (EM) method and the estimation algorithm is summarized in Table 1. The detailed derivation of the estimation method is given in Appendix A. The measurement model building algorithm is actually to alternate the evaluation of tij with fixed H and the update of H
pleft ¼
Nleft N
pcenter ¼ pright ¼
Ncenter N
Nright N
where Nleft ¼
N X
ti;left ðwi Þ;
Ncenter ¼
i¼1
N X i¼1
ti;center ðwi Þ;
Nright ¼
N X
ti;right ðwi Þ:
i¼1
4. Check the convergence criterion. If it is not met, go back to Step 2. If it is, return H ¼ fp; Hleft ; Hcenter ; Hright g.
with the fixed tij . The posterior probabilities tij are defined in Appendix A.
4. Target vehicle tracking 4.1. Vehicular model In the vehicle tracking, the motion of the target cannot be characterized at all times by a single model and a finite number of models called multiple models (MM) should be combined to adequately describe its motion in different regimes (Arulampalam, Maskell, Gordon, and Clapp, 2002; Li and Bar-Shalom, 1993; Ristic, Arulampalam, and Gordon, 2004). Two kinematic models are used to capture the motion of the target tracking. Each model is of third-order in two Cartesian coordinates on the horizontal driving plane and includes position, velocity and acceleration for each coordinate, leading to a total of six dimensions. The two kinematic models are represented by xk ¼ Fxk1 þ Guk
ð3Þ
where xk is the state vector of a preceding vehicle xk ¼ ½x x_ x€ Z Z_ Z€ T
ð4Þ
and x and Z denote the orthogonal coordinates of the horizontal plane. F and G are system and control input matrices, respectively,
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and they are defined by
the number of measurements from the microwave radar changes. This problem is considered case by case: (i) Case when the number of measurement equals 1: This is the basic case. Since the set in (7) defines the state and the mode of the preceding vehicle, the importance of each particle is weighted by ð5Þ wik pwik1
i pðzk jxik ; rki Þpðxik ; rki jxik1 ; rk1 Þ
¼ wik1 where T is the sampling time period. The matrices F and G can be decomposed into two independent block systems of x and Z as in (5) and are widely used for target tracking (Johnston and Krishnamurthy, 2001). uk is the process noise and denotes the acceleration increment during the kth sampling period. The two models differ only in their level of process noise variance. The low variance model ðrk ¼ 1Þ corresponds to a uniform motion while the high variance model ðrk ¼ 2Þ corresponds to a maneuvering motion, where rk denotes the mode of the multiple models. In the subsequent simulation, it is assumed that the sampling time T= 100 ms, 0 2 0 2 0:1umax 2umax Ru1 ¼ and Ru2 ¼ ; 0 0:2 0 0:2 where Ru1 and Ru2 are the covariance of uk for r = 1 and 2, respectively; max is the maximum of the acceleration increment applied to (3) to change a car lane within 3.2 s and 1 3:5 m 0:1 s umax ¼ ¼ 0:3418 m=s2 : 2 ð0:8s Þ2 0:8s The mode rk changes over time by a Markov chain with the transition kernel Kðrk1 ; rk Þ ¼ Pðrk ¼ ijrk1 ¼ jÞ ¼ Zij
ð6Þ
where
g ¼ ðZij Þ ¼
0:95 0:05
0:05 : 0:95
ð7Þ
where xik A R6 and rki A f1; 2g denote the state and the mode of the preceding vehicle, respectively, and wik denotes the associated importance weight. When rki ¼ 1, the preceding vehicle is in uniform mode and when rki ¼ 2, it is in maneuvering mode. Then, the posterior pðxk jZ1:k Þ is represented using Pk by pðxk ; rk jZ1:k Þ ¼
wik dðxk xik ; rk rki Þ
i ; zk Þ qðxik ; rki jxik1 ; rk1 i pðzk jxik ; rki Þpðxik jrki ; xik1 Þpðrki jrk1 Þ i ; zk Þ qðxik ; rki jxik1 ; rk1
:
ð9Þ
For the sake of simplicity, the proposal distribution is selected as i i ; zk Þ ¼ pðxik jrki ; xik1 Þpðrki jrk1 Þ qðxik ; rki jxik1 ; rk1
ð10Þ
and the particles are sampled from i i rki Pð djrk1 Þ ¼ Kðrk1 ; dÞ
ð11Þ
xik pð djrki ; xik1 Þ ¼ NðFxik1 ; Rw r i Þ:
ð12Þ
k
Then, the weight can be updated by wik pwik1 pðzk jxik ; rki Þ ¼ wik1 pðwik ¼ Hxik zk jxik ; rki Þ X pj pj ðwik jHj Þ; ¼ wik1
ð13Þ
j ¼ fleft;center;rightg
where the measurement (13) is used and k1i ; dÞ is the Markov transition kernel given in (6). (ii) Case when the number of measurement is equal to or greater than two: When the number of measurements is equal to or greater than two, the extra measurements besides the first one are regarded as next measurements involving no vehicular motion. For example, three measurements 1 zk , 2 zk and 3 zk are observed at time k, the weights are not updated by ð14Þ
i ; zk Þ ¼ Instead, each particle is drawn from qðxik ; rki jxik1 ; rk1 i Þ once and goes through the resampling three pðxik jrki ; xik1 Þpðrki jrk1 times using the following importance weights
The particle filter is a special version of the Bayes filter, and it is based on sequential Monte Carlo (SMC) sampling (Arulampalam et al., 2002; Ristic et al., 2004). As stated above, the hybrid particle filter is employed as a possible solution to this problem with complicated measurement properties. At the kth time, the posterior of the target and its mode is represented by a set of N particles
N X
i i pðzk jxik ; rki Þpðxik jrki ; xik1 ; rk1 Þpðrki jxik1 ; rk1 Þ
wik pwik1 pð1 zk ; 2 zk ; 3 zk jxik ; rki Þ:
4.2. Modified hybrid particle filter
Pk ¼ fðxik ; rki ; wik Þji ¼ 1; . . . ; Ng;
¼ wik1
i ; zk Þ qðxik ; rki jxik1 ; rk1
ð8Þ
1
wik pwik1 pð1 zk jxik ; rki Þ
ð15Þ
2
wik p1 wik pð2 zk jxik ; rki Þ
ð16Þ
wik p2 wik pð3 zk jxik ; rki Þ
ð17Þ
in turn. (iii) Case when no measurement is observed: In the case no observed measurement at k-th time, the particles are sampled by (11) and (12) but the importance weighting and resampling are skipped. Instead, at the ðk þ 1Þ-th time, the particle is drawn by xik þ 1 pð djrki þ 1 ; xik Þ ¼ NðFxk ; aRw ri
Þ
ð18Þ
kþ1
where a 41 to catch up to the target movement during the missing measurement. Then, the importance weights at the ðk þ 1Þ-th time are evaluated by
i¼1
where Z1:k denotes the observations accumulated up to k. A statistical measurement model (2) and multiple model dynamics (3) are employed in the hybrid particle filter to meet the requirements (1) and (2) in Section 2.2, respectively. Another difficulty still remains as to how to handle the situations in which
wik þ 1 pwik ¼ wik
pðzk þ 1 jxik þ 1 ; rki þ 1 Þpðxik þ 1 ; rki þ 1 jxik ; rki Þ qðxik þ 1 ; rki þ 1 jxik ; rki ; zk þ 1 Þ pðzk þ 1 jxik þ 1 ; rki þ 1 ÞNðFxik ; Rw ri
kþ1
Þpðrki þ 1 jrki Þ
NðFxik ; aRw Þpðrki þ 1 jrki Þ ri kþ1
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1 pwik pðzk þ 1 jxik þ 1 ; rki þ 1 Þexp ð1a1 Þðxik þ 1 Fxik ÞT 2 o w1 i i Rri ðxk þ 1 Fxk Þ
5. Simulation ð19Þ
kþ1
to reflect the change in the proposal distribution. The modified hybrid particle filter algorithm is summarized in Table 2. Table 2 The algorithm of a modified hybrid particle filter. N i i ½fxik ; rki gjN i ¼ 1 ¼ PF½fxk1 ; rk1 gji ¼ 1 ; zk ; zk1 For i ¼ 1 : N // Sampling i Draw rki Kðrk1 Þ 8 w if zk1 a + < NðFxik1 ; Rri Þ k Draw xik i : NðFxk1 ; aRw r i Þ if zk1 ¼ + k
If zk a +, For j ¼ 1 : cardðzk Þ 8 > > > > > > P > > > pp pp ðj wik jHp Þ > > < p ¼ fleft;center;rightg i if zk1 ¼ +; j ¼ 1 wk ¼ 1 > 1 i i > > exp ð1a1 Þðxik Fxik1 ÞT Rw > rki ðxk Fxk1 Þ > 2 > > > P > > > pp pp ðj wik jHp Þ otherwise :
In this section, some simulations are conducted to demonstrate the validity of the proposed method. The simulation uses real data from a microwave radar in order to emulate realistic tracking. First, a target vehicle is parked and the microwave radar is installed one meter away. The target vehicle and the radar are located in the same lane and a dataset Ssame with 1482 samples is built. Then, the target vehicle is located in the adjacent lane right to the radar and another dataset Sright with 452 samples is built. In turn, the vehicle is parked in the lane left to the radar and another dataset Sleft with 452 samples is built. Figs. 5(a)–(c) show the three datasets Ssame , Sright and Sleft , respectively. When the target vehicle is in the same lane with the radar, primarily left or right edges and occasionally in-between points are detected as shown in Fig. 5(a). When the vehicle is in the lane right to the radar, however, only the left edges are detected. When the vehicle is in the left lane, the right edges are detected. A separate measurement model is built for each case. A mixture model same same psame ðwjHsame Þ ¼ psame left Nðwjlleft ; Rleft Þ same same same same þ psame center UðwjAcenter Þ þ pright Nðwjlright ; Rright Þ
ð20Þ
p ¼ fleft;center;rightg
Resampling xik by wik //Resampling End End End Return fxik ; rki gjN i¼1
387
is built based on 1482 samples for the case of same lane, and two Gaussian models ; Rleft Þ pleft ðwjHÞ ¼ Nðwjlleft right right
ð21Þ
Fig. 5. Training data points from the target vehicles: (a) dataset Ssame from the same lane, (b) dataset Sright from the right lane, and (c) dataset Sleft from the left lane.
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results for scenarios 1–4, respectively. In the figures, the left plots correspond to the modified hybrid particle filter and the right plots correspond to the IMM–KF. In Figs. 8–11(a), the solid line denotes the actual movement of the maneuvering vehicle and ‘x’ denotes the measurement of the microwave radar. The measurement estimate of the vehicle is denoted as ‘o’, which is computed using the set of particles by N P
x^ k ¼
wik xik
i¼1 N P
i¼1
Fig. 6. Mixture probability with respect to the target position xpos;k .
and pright ðwjHÞ ¼ Nðwjlright ; Rright Þ left left
ð22Þ
are built for the case of adjacent lanes. It is assumed that the lane is 3.5 m wide, which is a Korean standard, and the radar measurements are emulated in the simulation by zk ¼ wk þ xpos;k 8 left > > > : Sright
ð23Þ with the probability b
left
with the probability b
same
with the probability b
right
where denotes the random sampling from the right side set; P l the mixture probabilities b satisfy bl ¼ 1 and are l ¼ fleft;same;rightg defined as in Fig. 6. The new mixture model left left
pðwjHÞ ¼ b
p
same same
ðwjHleft Þ þ b
p
right right
ðwjHsame Þ þ b
p
ðwjHright Þ
ð24Þ combining (20)–(22) is applied to the modified hybrid particle filter. It is assumed that the microwave radar is installed to the stop-and-go vehicle and the ICC car the radar is traveling along a straight three-lane road at a constant velocity of 10 m/s and the radar reliably measures up to 30 m. A maneuvering vehicle is preceding the ICC car at the velocity of 13 m/s. In the case of stopand-go, 10–15 m/s is a reasonable speed. The radar is queried every 100 ms and returns from zero to three detection points for the same target. The proposed method is applied to the following four scenarios shown in Fig. 7: Scenario 1: The maneuvering vehicle traveling at 13 m/s passes the ICC vehicle on its left. It cuts into the middle during 2.5 and 5.6 s and continues driving in the middle lane. Scenario 2: The maneuvering vehicle traveling at 13 m/s passes the ICC vehicle on its right this time. It cuts into the middle during 2.8 and 6.2 s and continues driving in the middle lane. Scenario 3: The maneuvering vehicle passes the ICC vehicle on its left. It cuts from the left lane into the right lane, changing two lanes back-to-back, during 2.2 and 8.5 s and continues driving in the right lane. Scenario 4: The preceding vehicle drives in the middle lane at 13 m/s. It tries to cut into the right lane around 2.4 s but the driver changes his (or her) mind and drives back to the middle lane around 5 s. In this simulation, the radar measurements are emulated by (23) and the modified hybrid particle filter with the radar model (24) and the IMM–KF (interacting multiple models–Kalman filter) are compared for the four scenarios. Figs. 8–11 show the tracking
ð25Þ wik
It is demonstrated for each of the four scenarios that the proposed method substantially outperforms the IMM–KF and the reason is that the well-developed statistical measurement model (24) is incorporated in the proposed tracking system. Figs. 8–10(b) show the mode transition and its estimate of the maneuvering vehicle. In the figures, the solid line denotes the actual ‘‘driving straight‘‘mode of the maneuvering vehicle while the dotted and solid dotted lines denote the estimates of ‘‘driving straight’’ and ‘‘changing lane’’ modes, respectively. The figures show that the driver’s intention is predicted reasonably well with slight delay in the proposed method but the prediction does not work in the IMM–KF. In Figs. 8–10(c), the number of measurements is depicted with respect to the simulation time. Only one measurement is detected for most of the time but occasionally zero or more than two measurements are collected, as shown in the figures. To check the reliability, thirty independent runs are made for both the proposed method and the IMM–KF. The root mean square of the estimation error over the 10 s vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 10 uT X fðx x^ k Þ2 þðZk Z^ k Þ2 g RMS ¼ t 10 k ¼ 0:T k
ð26Þ
are evaluated and they are summarized in Table 3, where T is the sampling time period, xk ¼ ½xk x_ k x€ k Zk Z_ k Z€ k T and ^ ^ x^ k ¼ ½x^ k x_ k x€ k Z^ k Z^_ k Z^€ k T . As stated, it can be seen that the proposed method substantially outperforms the IMM–KF in all cases for all the scenarios.
6. Conclusion In this paper, a probabilistic vehicle tracking method with 24 GHz microwave radar has been proposed for the ICC. In order to overcome the severe dispersion and noise issue of the microwave radar, the radar measurement data points were collected and a statistical model was built. Then a modified hybrid particle filter was proposed to cope with the missing or multiple measurements of the microwave radar. This proposed method was applied to the four driving scenarios and exhibited the excellent performance via a computer simulation. Only preceding vehicles of intermediate size are considered in this paper. The application of the proposed method to more general targets including the motor cycles, trucks, heavy vehicles or even pedestrians will be explored in future studies. In addition, as reported by Park et al. (2010) and Urazghildiiev et al. (2007), microwave radar may also potentially be utilized as a target identification sensor. Further research should to be conducted to investigate the possibility of simultaneous target tracking and identification via single microwave radar within a unified probabilistic framework.
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Fig. 7. Four scenarios for the simulation: (a) scenario 1, (b) scenario 2, (c) scenario 3, and (d) scenario 4.
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Fig. 8. The results of scenario 1. (a) Tracking results (left: proposed, right: IMM–KF). (b) Mode probabilities (left: proposed, right: IMM–KF). (c) The number of measurements using microwave radar (left: proposed, right: IMM–KF).
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391
Fig. 9. The results of scenario 2. (a) Tracking results (left: proposed, right: IMM–KF). (b) Mode probabilities (left: proposed, right: IMM–KF). (c) The number of measurements using microwave radar (left: proposed, right: IMM–KF).
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Fig. 10. The results of scenario 3. (a) Tracking results (left: proposed, right: IMM–KF). (b) Mode probabilities (left: proposed, right: IMM–KF). (c) The number of measurements using microwave radar (left: proposed, right: IMM–KF).
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Fig. 11. The results of scenario 4. (a) Tracking results (left: proposed, right: IMM–KF). (b) Mode probabilities (left: proposed, right: IMM–KF). (c) The number of measurements using microwave radar (left: proposed, right: IMM–KF).
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Table 3 Simulation results (RMS). Scenario 1
Mean Best Worst
Scenario 2
Scenario 3
Scenario 4
IMM-PF
Proposed
IMM-PF
Proposed
IMM-PF
Proposed
IMM-PF
Proposed
0.6471 0.6000 0.7008
0.0831 0.0675 0.0998
0.6810 0.6192 0.7283
0.0995 0.0698 0.5179
0.7851 0.7299 0.8427
0.0996 0.0747 0.2639
0.4453 0.3284 0.5924
0.0856 0.0668 0.1184
Acknowledgements This work was supported by Grant number R01-2006-00011016-0 from Basic Research Program of the Korea Science & Engineering Foundation.
where tij 9pðmi ¼ jjwi ; Hj Þ. Under the assumption that all the parameters Hleft , Hcenter and Hright are known, the posterior probability tij is computed by pðwi jmi ¼ j; Hj Þpðmi ¼ jjHj Þ P pðwi jmi ¼ k; Hk Þpðmi ¼ kjHk Þ
tij ¼ pðmi ¼ jjwi ; Hj Þ ¼
k ¼ fleft;center;rightg
Appendix A
p pj ðwi Þ pk pk ðwi Þ
Pj
¼
k ¼ fleft;center;rightg
and Z ¼ fzi gN Since the measurements X pos ¼ fxpos;i gN i ¼ 1 are i¼1 i.i.d, the likelihood can be rewritten into pðW; mjHÞ ¼
N Y
pðwi ; mi jHÞ ¼
i¼1
Y
pleft pleft ðwi jHleft Þ
i:mi ¼ left
Y
pcenter pcenter ðwi jHcenter Þ
i:mi ¼ center
Y
¼
Y
pright ðwi jHright Þ
Maximization step (M-step): In the M-step, the expected log P likelihood (A.3) is maximized subject to j ¼ fleft;center;rightg pj ¼ 1. The corresponding Lagrangian is
i:mi ¼ right
Y
pj pj ðwi jHj Þ
ðA:1Þ
Lðp; H; lÞ ¼
X
X
logðpj pj ðwi jHj ÞÞ
j ¼ fleft;center;rightgi:mi ¼ j
X
¼
N X
Iðmi ¼ jÞflog pj þ log pj ðwi jHj Þg
X
þ l@
¼
X
¼
N X i¼1 N X
ti;left
@ @lleft
n
o logNðwi jlleft ; Rleft Þ
D 1 log 2p log jRleft j 2 2 1 ðwi lleft ÞT R1 left ðwi lleft Þ 2 N X @ D 1 ti;left log 2p log jRleft j ¼ @lleft 2 2 i¼1 1 ðwi lleft ÞT R1 left ðwi lleft Þ 2 N X ti;left R1 ¼ left ðwi lleft Þ:
¼
ðA:2Þ
ti;left
i¼1
where IðdÞ is the indicator function and (A.2) is the function to be maximized. In reality, however, it is not known which part of the car is detected by the microwave radar and thus its expected value should be considered. Expectation step (E-step): Since mi is unknown for each data point i, the expected log likelihood over mi conditioned on the measurement wi is considered Emi ðlog pðW; mjHÞÞ ¼
ðA:5Þ
j ¼ fleft;center;rightg
Iðmi ¼ jÞflog pj þ log pj ðwi jHj Þg;
X
pj 1A
where l is a Lagrange multiplier. First, it is differentiated with respect to lleft and 2 N X @Lðp; H; lÞ @ 4X ¼ t flog pj þ log pj ðwi jHj Þg @lleft @lleft i ¼ 1 j ¼ fleft;center;rightg ij 0 13 X pj 1A5 þ l@
i ¼ 1 j ¼ fleft;center;rightg
N X
1
j ¼ fleft;center;rightg
j ¼ fleft;center;rightg i ¼ 1 N X
tij flog pj þlog pj ðwi jHj Þg
i ¼ 1 j ¼ fleft;center;rightg
0
j ¼ fleft;center;rightg i:mi ¼ j
¼
X
N X
j ¼ fleft;center;rightg i:mi ¼ j
where m ¼ fmi gN i ¼ 1 and mi denotes which part (left, center or right) of the preceding car is detected by the microwave radar at the i-th measurement and takes one of fleft; center; rightg. Here, the classical trick of taking logarithm is used. Thus, Y Y pj pj ðwi jHj Þ log pðW; mjHÞ ¼ log
ðA:4Þ
@
@lleft
ðA:6Þ
i¼1
Emi ðIðmi ¼ jÞÞflog pj
i ¼ 1 j ¼ fleft;center;rightg
¼
þ log pj ðwi jHj Þg N X X
By zeroing the derivative, pðmi ¼ jjwi ; Hj Þflog pj
PN
i ¼ 1 j ¼ fleft;center;rightg
¼
þ log pj ðwi jHj Þg N X X
lleft ¼
tij flog pj þlog pj ðwi jHj Þg;
i ¼ 1 j ¼ fleft;center;rightg
ðA:3Þ
N ti;left ðwi Þ 1 X ¼ t ðw Þ; Nleft i ¼ 1 i;left i i ¼ 1 ti;left
i¼1
PN
ðA:7Þ
P where Nleft ¼ N i ¼ 1 ti;left ðwi Þ. Then, the Lagrangian is differentiate it with respect to Rleft . For simplicity, Kleft ¼ R1 left is used instead
ARTICLE IN PRESS S. Park et al. / Control Engineering Practice 18 (2010) 383–395
of Sleft
2
N X
X @Lðp; H; lÞ @ 4 ¼ t flog pj þlog pj ðwi jHj Þg @Kleft @Kleft i ¼ 1 j ¼ fleft;center;rightg ij 0 13 X pj 1A5 þ l@ j ¼ fleft;center;rightg
¼
N X i¼1 N X
ti;left
@
n
@Kleft
log Nðwi jlleft ; K1 left Þ
o
D 1 log 2p þ logjKleft j 2 2
1 T ðwi lleft Þ Kleft ðwi mleft Þ 2 N X @ D 1 ti;left log 2p þ logjKleft j ¼ @Kleft 2 2 i¼1 X N 1 T ti;left ðK1T TrðKleft Si;left Þ ¼ left Si;left Þ; 2 i¼1 ¼
ti;left
i¼1
@
@Kleft
ðA:8Þ
where S i;left ðwi lleft Þðwi lleft ÞT . Zeroing the derivative yields PN N 1 X i ¼ 1 ti;left ðS i;left Þ Rleft ¼ K1 ¼ ¼ t ðw lleft Þðwi lleft ÞT : PN left Nleft i ¼ 1 i;left i i ¼ 1 ti;left ðA:9Þ The same steps are repeated for j ¼ right and N 1 X t ðw Þ Nright i ¼ 1 i;right i
lright ¼
Rright ¼
1
N X
Nright i ¼ 1
where Nright ¼ such that
ðA:10Þ
ti;right ðwi lright Þðwi lright ÞT ;
PN
i¼1
ðA:11Þ
ti;right ðwi Þ. For j ¼ center, Acenter is determined
fwi jti;center Z eg Acenter
ðA:12Þ
where e is a small positive number. Finally, the Lagrangian (A.5) is differentiated with respect to lleft by 2 N X @Lðp; H; kÞ @ 4X ¼ t flog pj þlog pj ðwi jHj Þg @pk @pk i ¼ 1 j ¼ fleft;center;rightg ij 0 13 N X X 1 N @ pj 1A5 ¼ tik þ l ¼ k þ l: ðA:13Þ þl j ¼ fleft;center;rightg
pk
i¼1
pk
By zeroing,
pk ¼
Nk
l
:
ðA:14Þ
Substituting it into (A.13) gives X X Nj N ¼ ¼ 1: pj ¼ j ¼ fleft;center;rightg
j ¼ fleft;center;rightg
l
l
ðA:15Þ
Thus,
pk ¼
Nk N
ðA:16Þ
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Contents lists available at ScienceDirect
Control Engineering Practice journal homepage: www.elsevier.com/locate/conengprac
Vehicle tracking using a microwave radar for situation awareness Seongkeun Park a, Jae Pil Hwang a, Euntai Kim a,, Hyung-Jin Kang b a b
School of Electrical and Electronic Engineering, Yonsei University, C613, Sinchon-dong, Seodaemun-gu, Seoul 120-749, Korea Mando Central Research Center, Gyeonggi-do 449-901, Korea
a r t i c l e in fo
abstract
Article history: Received 27 December 2008 Accepted 7 December 2009 Available online 18 January 2010
In this paper, a probabilistic target vehicle tracking method is proposed for situation awareness of intelligent cruise control (ICC) vehicle. The ICC vehicle considered herein is equipped with a 24 GHz microwave radar for tracking the preceding vehicle. To overcome the severe dispersion and noise of the microwave radar, a statistical model for the radar is built and it is applied to the hybrid particle filter. The hybrid particle filter is combined with the interacting multiple models (IMM) to track the preceding vehicle and predict the driver’s intention. Furthermore, the modified hybrid particle filter is proposed to cope with the missing or multiple measurements of the microwave radar. Finally, a computer simulation is conducted and the validity of the proposed method is demonstrated. & 2009 Elsevier Ltd. All rights reserved.
Keywords: Microwave radar Scattering IMM Hybrid particle filter EM
1. Introduction The majority of automobile companies have recently shifted their research focus from the classical passive safety system such as safety belt (Chan, 2007; Watanabe, Umezawa, and Abe, 1994) to various active driver assistance systems. In particular, intelligent cruise control (ICC) including adaptive cruise control (ACC) and stop-and-go has received considerable attention within the vehicular community with the recent advances of monolithic microwave integrated circuit (MMIC), integrated circuit (IC) (Naranjo, Labayrade, Royere, Hautiere, and Aubert, 2007; Russell et al., 1997; Tsang, Hall, Hoare, and Clarke, 2006) and computers (Alessandretti, Broggi, and Cerri, 2007;Forsyth and Ponce, 2002; ¨ Holzmann, Halfmann, Germann, Wurtenberger, and Isermann, 1997;Hwang, Rou, Park, Kim, and Kang, 2006;Moon, Moon, and Yi, 2009;Perrollaz et al., 2006). Fig. 1 shows the configuration of an ICC system (Kim and Hong, 2004). The performance of an ICC system depends on how accurate the sensors are in recognizing traffic and obstacles ahead. The sensor candidates include a lidar (Guang and Tomizuka, 2003; Takagi, Morikawa, Ogawa, and Saburi, 2006), a vision (Sotelo et al., 2004; Stein, Mano, and Shashua 2003) and a radar (Eriksson and As, 1997; Moon and Yi, 2002). When compared with the lidar, the microwave radar has the following properties: (1) Microwave radar is more robust in all weather conditions including fog and rain. (2) Microwave radar can measure the relative velocities of the nearby objects using Dopplers’ effect (Russell et al., 1997) and Corresponding author.
E-mail address: [email protected] (E. Kim). 0967-0661/$ - see front matter & 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.conengprac.2009.12.006
therefore is widely used as a range sensor in most of the commercial ICC systems. (3) In general, the price of radar used to be much higher than that of lidar. But with the rapid growth of the silicon semiconductor technology, the radar can be manufactured not only using chemical semiconductor such as GaAs (Larson, Hackett, and Lohr, 1991) but also using silicon semiconductor such as SiGe (Bock et al., 2004). Thus, the price competitiveness of the radar gets improved against that of the lidar these days. Thus, today the price of radar is only slightly more expensive than that of lidar. (4) The radar can be installed behind a vehicle bumper or other cover, but the lidar must be installed in an exposed area. Thus, radar is preferred to lidar from the perspective of design. (5) The relative velocities of preceding vehicles can be measured directly by radar. Lidar, however, cannot measure relative velocity, which therefore must be computed by differentiating relative distances. When compared with the vision, the microwave radar has the following properties: (1) In general, the vision cameras are cheaper than microwave radar and are still capable of returning rich environmental information. (2) However, the vision is much less robust in the context of light and weather variations, and range measurement is inaccurate. (3) Information from the vision cameras is so rich that tremendous computational burdens are required to localize vehicles within images. (4) Consequently, the vision cameras are recommended only as secondary sensors to millimeter radars.
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Fig. 1. The configuration of an ICC system.
In summary, microwave radar is a key sensor in the ICC at present and will continue to remain so in the future. Some previous works have reported the studies about vehicle tracking using a microwave radar (Gresham et al., 2004; Kim and Hong, 2004; Stursberg, Fehnker, Han, and Krogh, 2004; Venhovens, Naab, and Adiprasito, 2000). To my knowledge, however, most of the previous works treated the radar as a sensor with Gaussian noise, and none of them fully considered the dispersive and noisy nature of the microwave radar when it is used for target tracking. In this paper, a target vehicle tracking method is presented for the ICC using a microwave radar. To deal with the complicated characteristics of the microwave radar, a statistical model of the radar is built and it is combined with the hybrid particle filter to track the target vehicle. This paper is a companion paper of Park, Hwang, Kim, Lee, and Jung work (2010), in which the same radar was used for target identification. The remainder of this paper is organized as follows: In Section 2, microwave radar used herein is introduced and the problem is described in detail along with the solution requirements In Section 3, the radar measurements are statistically modeled. In Section 4, a modified hybrid particle filter is proposed to estimate the state of the preceding vehicle. In Section 5, simulation is conducted. Finally, some conclusions are drawn in Section 6.
Fig. 2. The ICC vehicle equipped with a 24 GHz microwave radar.
Fig. 3. A driving situation.
2. Problem formulation 2.1. Microwave radar and adaptive cruise control The microwave radar used in this paper, MASRAU0025, is manufactured by M/A-COM and is composed of a 24 GHz radar sensor and microprocessor unit (Manual of MA-COM MASRAU0025). The MASRAU0025 detects the objects up to 30 m ahead and returns a variety of information about the object including the distance, azimuth angle, velocity, etc. The ICC vehicle is equipped with the microwave radar MASRAU0025, as shown in Fig. 2. As shown in figure, MASRAU0025 is mounted on the car and it is connected to a laptop computer through a control area network (CAN). The laptop computer has a P-IV processor and the program is operating on Windows XP. Fig. 3 shows the driving situation considered in this paper in which a target vehicle and the ICC vehicle are traveling on a three-lane road. The target vehicle changes the lane in a potentially dangerous manner and passes the ICC. The ICC should track the preceding target vehicle and identify the intention of the driver via the microwave radar. 2.2. Problem description and solution Here, the vehicle tracking problem will now be described in detail.
Fig. 4. Points of a vehicle detected by a 24 GHz radar.
(1) The microwave radar easily disperses and the detection points on the vehicle that the radar detects will vary every time. Mainly, the radar detects the left corner or right corner of the vehicle ahead, but occasionally, the radar detects points in between, as shown in Fig. 4. When the radar detects the
ARTICLE IN PRESS S. Park et al. / Control Engineering Practice 18 (2010) 383–395
in-between points, the measured point is almost random between the two corner points. (2) For a single scan, the number of measurements that the microwave radar may return for a target vehicle varies from zero to three. With the chance of over 80%, the radar returns a single measurement. Occasionally, however, the radar returns two or three measurements for a target or does not return at all. (3) The intention of the driver should be predicted from the movement of the preceding target vehicle, for example, whether the vehicle is moving uniformly or changing lanes. This vehicle tracking problem does not allow for typical solutions due to the above difficulties. More specifically, the Kalman filter, or its variants, cannot be applied due to the difficulties described in the radar and combine it with the particle filter (PF). To meet requirement (3), the PF is combined with the interacting multiple model (IMM) to predict the driver’s intention. The result is a hybrid particle filter with both continuous and discrete states.
In this section, a statistical model is build for the microwave radar MASRAU0025. The statistical model is utilized in tracking a preceding target vehicle in a probabilistic way. As stated, MASRAU0025 detects the left corner or right corner of the vehicle, or the middle part in between, mostly more than once. and Z ¼ fzi gN First, two sets of data X pos ¼ fxpos;i gN i ¼ 1 are i¼1 collected using the MASRAU0025, where xpos;i denotes the true position of a car and zi denotes the associated measurement that microwave radar returns. Then, the measurement error is defined as ð1Þ
and its statistical model is built from the data set W ¼ fwi gN i ¼ 1. The measurements from the radar are modeled using a mixture of three components and the parameters of the model are identified by the Expectation and Maximization (EM) (Bishop, 2006). The mixture is composed of two Gaussians for the left and right edges and a uniform for the center face in between. The measurement model is X pj pj ðwjHj Þ pðwjHÞ ¼ j ¼ fleft;center;rightg
¼ pleft Nðwjlleft ; Rleft Þ þ pcenter UðwjAcenter Þ þ pright Nðwjlright ; Rright Þ
Table 1 The algorithm of building vehicle measurement model. Algorithm building measurement model ðWÞ 1. Initialize H ¼ fp; Hleft ; Hcenter ; Hright g 2. For all wi , assign the data to each component by
pj pj ðwi Þ
tij ¼ P
k ¼ fleft;center;rightg
pk pk ðwi Þ
3. Update the parameters by
lleft ¼
N 1 X t ðw Þ Nleft i ¼ 1 i;left i
Rleft ¼
N 1 X t ðw lleft Þðwi lleft ÞT Nleft i ¼ 1 i;left i
lright ¼
N 1 X t ðw Þ Nright i ¼ 1 i;right i
Rright ¼
N 1 X t ðw lright Þðwi lright ÞT Nright i ¼ 1 i;right i
Acenter *fwi jti;center Z eg
3. Measurement model of microwave radar
wi ¼ zi xpos;i
385
ð2Þ
where pleft ðÞ and rightðÞ are the Gaussian distributions NðÞ and pcenter ðÞ is the uniform distribution UðÞ. pleft ¼ pðj ¼ leftÞ is a prior probability that this detection is from the left edge and pcenter and pright are defined in the similar way. Further, P j ¼ fleft;center;rightg pj ¼ 1. pleft ð dÞ is a distribution of w when the microwave radar detects the left edge of the preceding car and centerðÞ and pright ðÞ are defined in the similar way. Further H ¼ fp; Hleft ; A Hcenter ; Hright g, p ¼ fpleft ; pcenter ; pright g, Hcenter ¼ ðAcenter Þ, Hright ¼ ðlright ; Rright Þ; lj and Rj are the mean and covariance of the Gaussians, respectively, and Acenter is the area on which measurements are detected uniformly. The parameters of the measurement model can be estimated by the Expectation–Maximization (EM) method and the estimation algorithm is summarized in Table 1. The detailed derivation of the estimation method is given in Appendix A. The measurement model building algorithm is actually to alternate the evaluation of tij with fixed H and the update of H
pleft ¼
Nleft N
pcenter ¼ pright ¼
Ncenter N
Nright N
where Nleft ¼
N X
ti;left ðwi Þ;
Ncenter ¼
i¼1
N X i¼1
ti;center ðwi Þ;
Nright ¼
N X
ti;right ðwi Þ:
i¼1
4. Check the convergence criterion. If it is not met, go back to Step 2. If it is, return H ¼ fp; Hleft ; Hcenter ; Hright g.
with the fixed tij . The posterior probabilities tij are defined in Appendix A.
4. Target vehicle tracking 4.1. Vehicular model In the vehicle tracking, the motion of the target cannot be characterized at all times by a single model and a finite number of models called multiple models (MM) should be combined to adequately describe its motion in different regimes (Arulampalam, Maskell, Gordon, and Clapp, 2002; Li and Bar-Shalom, 1993; Ristic, Arulampalam, and Gordon, 2004). Two kinematic models are used to capture the motion of the target tracking. Each model is of third-order in two Cartesian coordinates on the horizontal driving plane and includes position, velocity and acceleration for each coordinate, leading to a total of six dimensions. The two kinematic models are represented by xk ¼ Fxk1 þ Guk
ð3Þ
where xk is the state vector of a preceding vehicle xk ¼ ½x x_ x€ Z Z_ Z€ T
ð4Þ
and x and Z denote the orthogonal coordinates of the horizontal plane. F and G are system and control input matrices, respectively,
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and they are defined by
the number of measurements from the microwave radar changes. This problem is considered case by case: (i) Case when the number of measurement equals 1: This is the basic case. Since the set in (7) defines the state and the mode of the preceding vehicle, the importance of each particle is weighted by ð5Þ wik pwik1
i pðzk jxik ; rki Þpðxik ; rki jxik1 ; rk1 Þ
¼ wik1 where T is the sampling time period. The matrices F and G can be decomposed into two independent block systems of x and Z as in (5) and are widely used for target tracking (Johnston and Krishnamurthy, 2001). uk is the process noise and denotes the acceleration increment during the kth sampling period. The two models differ only in their level of process noise variance. The low variance model ðrk ¼ 1Þ corresponds to a uniform motion while the high variance model ðrk ¼ 2Þ corresponds to a maneuvering motion, where rk denotes the mode of the multiple models. In the subsequent simulation, it is assumed that the sampling time T= 100 ms, 0 2 0 2 0:1umax 2umax Ru1 ¼ and Ru2 ¼ ; 0 0:2 0 0:2 where Ru1 and Ru2 are the covariance of uk for r = 1 and 2, respectively; max is the maximum of the acceleration increment applied to (3) to change a car lane within 3.2 s and 1 3:5 m 0:1 s umax ¼ ¼ 0:3418 m=s2 : 2 ð0:8s Þ2 0:8s The mode rk changes over time by a Markov chain with the transition kernel Kðrk1 ; rk Þ ¼ Pðrk ¼ ijrk1 ¼ jÞ ¼ Zij
ð6Þ
where
g ¼ ðZij Þ ¼
0:95 0:05
0:05 : 0:95
ð7Þ
where xik A R6 and rki A f1; 2g denote the state and the mode of the preceding vehicle, respectively, and wik denotes the associated importance weight. When rki ¼ 1, the preceding vehicle is in uniform mode and when rki ¼ 2, it is in maneuvering mode. Then, the posterior pðxk jZ1:k Þ is represented using Pk by pðxk ; rk jZ1:k Þ ¼
wik dðxk xik ; rk rki Þ
i ; zk Þ qðxik ; rki jxik1 ; rk1 i pðzk jxik ; rki Þpðxik jrki ; xik1 Þpðrki jrk1 Þ i ; zk Þ qðxik ; rki jxik1 ; rk1
:
ð9Þ
For the sake of simplicity, the proposal distribution is selected as i i ; zk Þ ¼ pðxik jrki ; xik1 Þpðrki jrk1 Þ qðxik ; rki jxik1 ; rk1
ð10Þ
and the particles are sampled from i i rki Pð djrk1 Þ ¼ Kðrk1 ; dÞ
ð11Þ
xik pð djrki ; xik1 Þ ¼ NðFxik1 ; Rw r i Þ:
ð12Þ
k
Then, the weight can be updated by wik pwik1 pðzk jxik ; rki Þ ¼ wik1 pðwik ¼ Hxik zk jxik ; rki Þ X pj pj ðwik jHj Þ; ¼ wik1
ð13Þ
j ¼ fleft;center;rightg
where the measurement (13) is used and k1i ; dÞ is the Markov transition kernel given in (6). (ii) Case when the number of measurement is equal to or greater than two: When the number of measurements is equal to or greater than two, the extra measurements besides the first one are regarded as next measurements involving no vehicular motion. For example, three measurements 1 zk , 2 zk and 3 zk are observed at time k, the weights are not updated by ð14Þ
i ; zk Þ ¼ Instead, each particle is drawn from qðxik ; rki jxik1 ; rk1 i Þ once and goes through the resampling three pðxik jrki ; xik1 Þpðrki jrk1 times using the following importance weights
The particle filter is a special version of the Bayes filter, and it is based on sequential Monte Carlo (SMC) sampling (Arulampalam et al., 2002; Ristic et al., 2004). As stated above, the hybrid particle filter is employed as a possible solution to this problem with complicated measurement properties. At the kth time, the posterior of the target and its mode is represented by a set of N particles
N X
i i pðzk jxik ; rki Þpðxik jrki ; xik1 ; rk1 Þpðrki jxik1 ; rk1 Þ
wik pwik1 pð1 zk ; 2 zk ; 3 zk jxik ; rki Þ:
4.2. Modified hybrid particle filter
Pk ¼ fðxik ; rki ; wik Þji ¼ 1; . . . ; Ng;
¼ wik1
i ; zk Þ qðxik ; rki jxik1 ; rk1
ð8Þ
1
wik pwik1 pð1 zk jxik ; rki Þ
ð15Þ
2
wik p1 wik pð2 zk jxik ; rki Þ
ð16Þ
wik p2 wik pð3 zk jxik ; rki Þ
ð17Þ
in turn. (iii) Case when no measurement is observed: In the case no observed measurement at k-th time, the particles are sampled by (11) and (12) but the importance weighting and resampling are skipped. Instead, at the ðk þ 1Þ-th time, the particle is drawn by xik þ 1 pð djrki þ 1 ; xik Þ ¼ NðFxk ; aRw ri
Þ
ð18Þ
kþ1
where a 41 to catch up to the target movement during the missing measurement. Then, the importance weights at the ðk þ 1Þ-th time are evaluated by
i¼1
where Z1:k denotes the observations accumulated up to k. A statistical measurement model (2) and multiple model dynamics (3) are employed in the hybrid particle filter to meet the requirements (1) and (2) in Section 2.2, respectively. Another difficulty still remains as to how to handle the situations in which
wik þ 1 pwik ¼ wik
pðzk þ 1 jxik þ 1 ; rki þ 1 Þpðxik þ 1 ; rki þ 1 jxik ; rki Þ qðxik þ 1 ; rki þ 1 jxik ; rki ; zk þ 1 Þ pðzk þ 1 jxik þ 1 ; rki þ 1 ÞNðFxik ; Rw ri
kþ1
Þpðrki þ 1 jrki Þ
NðFxik ; aRw Þpðrki þ 1 jrki Þ ri kþ1
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1 pwik pðzk þ 1 jxik þ 1 ; rki þ 1 Þexp ð1a1 Þðxik þ 1 Fxik ÞT 2 o w1 i i Rri ðxk þ 1 Fxk Þ
5. Simulation ð19Þ
kþ1
to reflect the change in the proposal distribution. The modified hybrid particle filter algorithm is summarized in Table 2. Table 2 The algorithm of a modified hybrid particle filter. N i i ½fxik ; rki gjN i ¼ 1 ¼ PF½fxk1 ; rk1 gji ¼ 1 ; zk ; zk1 For i ¼ 1 : N // Sampling i Draw rki Kðrk1 Þ 8 w if zk1 a + < NðFxik1 ; Rri Þ k Draw xik i : NðFxk1 ; aRw r i Þ if zk1 ¼ + k
If zk a +, For j ¼ 1 : cardðzk Þ 8 > > > > > > P > > > pp pp ðj wik jHp Þ > > < p ¼ fleft;center;rightg i if zk1 ¼ +; j ¼ 1 wk ¼ 1 > 1 i i > > exp ð1a1 Þðxik Fxik1 ÞT Rw > rki ðxk Fxk1 Þ > 2 > > > P > > > pp pp ðj wik jHp Þ otherwise :
In this section, some simulations are conducted to demonstrate the validity of the proposed method. The simulation uses real data from a microwave radar in order to emulate realistic tracking. First, a target vehicle is parked and the microwave radar is installed one meter away. The target vehicle and the radar are located in the same lane and a dataset Ssame with 1482 samples is built. Then, the target vehicle is located in the adjacent lane right to the radar and another dataset Sright with 452 samples is built. In turn, the vehicle is parked in the lane left to the radar and another dataset Sleft with 452 samples is built. Figs. 5(a)–(c) show the three datasets Ssame , Sright and Sleft , respectively. When the target vehicle is in the same lane with the radar, primarily left or right edges and occasionally in-between points are detected as shown in Fig. 5(a). When the vehicle is in the lane right to the radar, however, only the left edges are detected. When the vehicle is in the left lane, the right edges are detected. A separate measurement model is built for each case. A mixture model same same psame ðwjHsame Þ ¼ psame left Nðwjlleft ; Rleft Þ same same same same þ psame center UðwjAcenter Þ þ pright Nðwjlright ; Rright Þ
ð20Þ
p ¼ fleft;center;rightg
Resampling xik by wik //Resampling End End End Return fxik ; rki gjN i¼1
387
is built based on 1482 samples for the case of same lane, and two Gaussian models ; Rleft Þ pleft ðwjHÞ ¼ Nðwjlleft right right
ð21Þ
Fig. 5. Training data points from the target vehicles: (a) dataset Ssame from the same lane, (b) dataset Sright from the right lane, and (c) dataset Sleft from the left lane.
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results for scenarios 1–4, respectively. In the figures, the left plots correspond to the modified hybrid particle filter and the right plots correspond to the IMM–KF. In Figs. 8–11(a), the solid line denotes the actual movement of the maneuvering vehicle and ‘x’ denotes the measurement of the microwave radar. The measurement estimate of the vehicle is denoted as ‘o’, which is computed using the set of particles by N P
x^ k ¼
wik xik
i¼1 N P
i¼1
Fig. 6. Mixture probability with respect to the target position xpos;k .
and pright ðwjHÞ ¼ Nðwjlright ; Rright Þ left left
ð22Þ
are built for the case of adjacent lanes. It is assumed that the lane is 3.5 m wide, which is a Korean standard, and the radar measurements are emulated in the simulation by zk ¼ wk þ xpos;k 8 left > >
ð23Þ with the probability b
left
with the probability b
same
with the probability b
right
where denotes the random sampling from the right side set; P l the mixture probabilities b satisfy bl ¼ 1 and are l ¼ fleft;same;rightg defined as in Fig. 6. The new mixture model left left
pðwjHÞ ¼ b
p
same same
ðwjHleft Þ þ b
p
right right
ðwjHsame Þ þ b
p
ðwjHright Þ
ð24Þ combining (20)–(22) is applied to the modified hybrid particle filter. It is assumed that the microwave radar is installed to the stop-and-go vehicle and the ICC car the radar is traveling along a straight three-lane road at a constant velocity of 10 m/s and the radar reliably measures up to 30 m. A maneuvering vehicle is preceding the ICC car at the velocity of 13 m/s. In the case of stopand-go, 10–15 m/s is a reasonable speed. The radar is queried every 100 ms and returns from zero to three detection points for the same target. The proposed method is applied to the following four scenarios shown in Fig. 7: Scenario 1: The maneuvering vehicle traveling at 13 m/s passes the ICC vehicle on its left. It cuts into the middle during 2.5 and 5.6 s and continues driving in the middle lane. Scenario 2: The maneuvering vehicle traveling at 13 m/s passes the ICC vehicle on its right this time. It cuts into the middle during 2.8 and 6.2 s and continues driving in the middle lane. Scenario 3: The maneuvering vehicle passes the ICC vehicle on its left. It cuts from the left lane into the right lane, changing two lanes back-to-back, during 2.2 and 8.5 s and continues driving in the right lane. Scenario 4: The preceding vehicle drives in the middle lane at 13 m/s. It tries to cut into the right lane around 2.4 s but the driver changes his (or her) mind and drives back to the middle lane around 5 s. In this simulation, the radar measurements are emulated by (23) and the modified hybrid particle filter with the radar model (24) and the IMM–KF (interacting multiple models–Kalman filter) are compared for the four scenarios. Figs. 8–11 show the tracking
ð25Þ wik
It is demonstrated for each of the four scenarios that the proposed method substantially outperforms the IMM–KF and the reason is that the well-developed statistical measurement model (24) is incorporated in the proposed tracking system. Figs. 8–10(b) show the mode transition and its estimate of the maneuvering vehicle. In the figures, the solid line denotes the actual ‘‘driving straight‘‘mode of the maneuvering vehicle while the dotted and solid dotted lines denote the estimates of ‘‘driving straight’’ and ‘‘changing lane’’ modes, respectively. The figures show that the driver’s intention is predicted reasonably well with slight delay in the proposed method but the prediction does not work in the IMM–KF. In Figs. 8–10(c), the number of measurements is depicted with respect to the simulation time. Only one measurement is detected for most of the time but occasionally zero or more than two measurements are collected, as shown in the figures. To check the reliability, thirty independent runs are made for both the proposed method and the IMM–KF. The root mean square of the estimation error over the 10 s vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 10 uT X fðx x^ k Þ2 þðZk Z^ k Þ2 g RMS ¼ t 10 k ¼ 0:T k
ð26Þ
are evaluated and they are summarized in Table 3, where T is the sampling time period, xk ¼ ½xk x_ k x€ k Zk Z_ k Z€ k T and ^ ^ x^ k ¼ ½x^ k x_ k x€ k Z^ k Z^_ k Z^€ k T . As stated, it can be seen that the proposed method substantially outperforms the IMM–KF in all cases for all the scenarios.
6. Conclusion In this paper, a probabilistic vehicle tracking method with 24 GHz microwave radar has been proposed for the ICC. In order to overcome the severe dispersion and noise issue of the microwave radar, the radar measurement data points were collected and a statistical model was built. Then a modified hybrid particle filter was proposed to cope with the missing or multiple measurements of the microwave radar. This proposed method was applied to the four driving scenarios and exhibited the excellent performance via a computer simulation. Only preceding vehicles of intermediate size are considered in this paper. The application of the proposed method to more general targets including the motor cycles, trucks, heavy vehicles or even pedestrians will be explored in future studies. In addition, as reported by Park et al. (2010) and Urazghildiiev et al. (2007), microwave radar may also potentially be utilized as a target identification sensor. Further research should to be conducted to investigate the possibility of simultaneous target tracking and identification via single microwave radar within a unified probabilistic framework.
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Fig. 7. Four scenarios for the simulation: (a) scenario 1, (b) scenario 2, (c) scenario 3, and (d) scenario 4.
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Fig. 8. The results of scenario 1. (a) Tracking results (left: proposed, right: IMM–KF). (b) Mode probabilities (left: proposed, right: IMM–KF). (c) The number of measurements using microwave radar (left: proposed, right: IMM–KF).
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391
Fig. 9. The results of scenario 2. (a) Tracking results (left: proposed, right: IMM–KF). (b) Mode probabilities (left: proposed, right: IMM–KF). (c) The number of measurements using microwave radar (left: proposed, right: IMM–KF).
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Fig. 10. The results of scenario 3. (a) Tracking results (left: proposed, right: IMM–KF). (b) Mode probabilities (left: proposed, right: IMM–KF). (c) The number of measurements using microwave radar (left: proposed, right: IMM–KF).
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393
Fig. 11. The results of scenario 4. (a) Tracking results (left: proposed, right: IMM–KF). (b) Mode probabilities (left: proposed, right: IMM–KF). (c) The number of measurements using microwave radar (left: proposed, right: IMM–KF).
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Table 3 Simulation results (RMS). Scenario 1
Mean Best Worst
Scenario 2
Scenario 3
Scenario 4
IMM-PF
Proposed
IMM-PF
Proposed
IMM-PF
Proposed
IMM-PF
Proposed
0.6471 0.6000 0.7008
0.0831 0.0675 0.0998
0.6810 0.6192 0.7283
0.0995 0.0698 0.5179
0.7851 0.7299 0.8427
0.0996 0.0747 0.2639
0.4453 0.3284 0.5924
0.0856 0.0668 0.1184
Acknowledgements This work was supported by Grant number R01-2006-00011016-0 from Basic Research Program of the Korea Science & Engineering Foundation.
where tij 9pðmi ¼ jjwi ; Hj Þ. Under the assumption that all the parameters Hleft , Hcenter and Hright are known, the posterior probability tij is computed by pðwi jmi ¼ j; Hj Þpðmi ¼ jjHj Þ P pðwi jmi ¼ k; Hk Þpðmi ¼ kjHk Þ
tij ¼ pðmi ¼ jjwi ; Hj Þ ¼
k ¼ fleft;center;rightg
Appendix A
p pj ðwi Þ pk pk ðwi Þ
Pj
¼
k ¼ fleft;center;rightg
and Z ¼ fzi gN Since the measurements X pos ¼ fxpos;i gN i ¼ 1 are i¼1 i.i.d, the likelihood can be rewritten into pðW; mjHÞ ¼
N Y
pðwi ; mi jHÞ ¼
i¼1
Y
pleft pleft ðwi jHleft Þ
i:mi ¼ left
Y
pcenter pcenter ðwi jHcenter Þ
i:mi ¼ center
Y
¼
Y
pright ðwi jHright Þ
Maximization step (M-step): In the M-step, the expected log P likelihood (A.3) is maximized subject to j ¼ fleft;center;rightg pj ¼ 1. The corresponding Lagrangian is
i:mi ¼ right
Y
pj pj ðwi jHj Þ
ðA:1Þ
Lðp; H; lÞ ¼
X
X
logðpj pj ðwi jHj ÞÞ
j ¼ fleft;center;rightgi:mi ¼ j
X
¼
N X
Iðmi ¼ jÞflog pj þ log pj ðwi jHj Þg
X
þ l@
¼
X
¼
N X i¼1 N X
ti;left
@ @lleft
n
o logNðwi jlleft ; Rleft Þ
D 1 log 2p log jRleft j 2 2 1 ðwi lleft ÞT R1 left ðwi lleft Þ 2 N X @ D 1 ti;left log 2p log jRleft j ¼ @lleft 2 2 i¼1 1 ðwi lleft ÞT R1 left ðwi lleft Þ 2 N X ti;left R1 ¼ left ðwi lleft Þ:
¼
ðA:2Þ
ti;left
i¼1
where IðdÞ is the indicator function and (A.2) is the function to be maximized. In reality, however, it is not known which part of the car is detected by the microwave radar and thus its expected value should be considered. Expectation step (E-step): Since mi is unknown for each data point i, the expected log likelihood over mi conditioned on the measurement wi is considered Emi ðlog pðW; mjHÞÞ ¼
ðA:5Þ
j ¼ fleft;center;rightg
Iðmi ¼ jÞflog pj þ log pj ðwi jHj Þg;
X
pj 1A
where l is a Lagrange multiplier. First, it is differentiated with respect to lleft and 2 N X @Lðp; H; lÞ @ 4X ¼ t flog pj þ log pj ðwi jHj Þg @lleft @lleft i ¼ 1 j ¼ fleft;center;rightg ij 0 13 X pj 1A5 þ l@
i ¼ 1 j ¼ fleft;center;rightg
N X
1
j ¼ fleft;center;rightg
j ¼ fleft;center;rightg i ¼ 1 N X
tij flog pj þlog pj ðwi jHj Þg
i ¼ 1 j ¼ fleft;center;rightg
0
j ¼ fleft;center;rightg i:mi ¼ j
¼
X
N X
j ¼ fleft;center;rightg i:mi ¼ j
where m ¼ fmi gN i ¼ 1 and mi denotes which part (left, center or right) of the preceding car is detected by the microwave radar at the i-th measurement and takes one of fleft; center; rightg. Here, the classical trick of taking logarithm is used. Thus, Y Y pj pj ðwi jHj Þ log pðW; mjHÞ ¼ log
ðA:4Þ
@
@lleft
ðA:6Þ
i¼1
Emi ðIðmi ¼ jÞÞflog pj
i ¼ 1 j ¼ fleft;center;rightg
¼
þ log pj ðwi jHj Þg N X X
By zeroing the derivative, pðmi ¼ jjwi ; Hj Þflog pj
PN
i ¼ 1 j ¼ fleft;center;rightg
¼
þ log pj ðwi jHj Þg N X X
lleft ¼
tij flog pj þlog pj ðwi jHj Þg;
i ¼ 1 j ¼ fleft;center;rightg
ðA:3Þ
N ti;left ðwi Þ 1 X ¼ t ðw Þ; Nleft i ¼ 1 i;left i i ¼ 1 ti;left
i¼1
PN
ðA:7Þ
P where Nleft ¼ N i ¼ 1 ti;left ðwi Þ. Then, the Lagrangian is differentiate it with respect to Rleft . For simplicity, Kleft ¼ R1 left is used instead
ARTICLE IN PRESS S. Park et al. / Control Engineering Practice 18 (2010) 383–395
of Sleft
2
N X
X @Lðp; H; lÞ @ 4 ¼ t flog pj þlog pj ðwi jHj Þg @Kleft @Kleft i ¼ 1 j ¼ fleft;center;rightg ij 0 13 X pj 1A5 þ l@ j ¼ fleft;center;rightg
¼
N X i¼1 N X
ti;left
@
n
@Kleft
log Nðwi jlleft ; K1 left Þ
o
D 1 log 2p þ logjKleft j 2 2
1 T ðwi lleft Þ Kleft ðwi mleft Þ 2 N X @ D 1 ti;left log 2p þ logjKleft j ¼ @Kleft 2 2 i¼1 X N 1 T ti;left ðK1T TrðKleft Si;left Þ ¼ left Si;left Þ; 2 i¼1 ¼
ti;left
i¼1
@
@Kleft
ðA:8Þ
where S i;left ðwi lleft Þðwi lleft ÞT . Zeroing the derivative yields PN N 1 X i ¼ 1 ti;left ðS i;left Þ Rleft ¼ K1 ¼ ¼ t ðw lleft Þðwi lleft ÞT : PN left Nleft i ¼ 1 i;left i i ¼ 1 ti;left ðA:9Þ The same steps are repeated for j ¼ right and N 1 X t ðw Þ Nright i ¼ 1 i;right i
lright ¼
Rright ¼
1
N X
Nright i ¼ 1
where Nright ¼ such that
ðA:10Þ
ti;right ðwi lright Þðwi lright ÞT ;
PN
i¼1
ðA:11Þ
ti;right ðwi Þ. For j ¼ center, Acenter is determined
fwi jti;center Z eg Acenter
ðA:12Þ
where e is a small positive number. Finally, the Lagrangian (A.5) is differentiated with respect to lleft by 2 N X @Lðp; H; kÞ @ 4X ¼ t flog pj þlog pj ðwi jHj Þg @pk @pk i ¼ 1 j ¼ fleft;center;rightg ij 0 13 N X X 1 N @ pj 1A5 ¼ tik þ l ¼ k þ l: ðA:13Þ þl j ¼ fleft;center;rightg
pk
i¼1
pk
By zeroing,
pk ¼
Nk
l
:
ðA:14Þ
Substituting it into (A.13) gives X X Nj N ¼ ¼ 1: pj ¼ j ¼ fleft;center;rightg
j ¼ fleft;center;rightg
l
l
ðA:15Þ
Thus,
pk ¼
Nk N
ðA:16Þ
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