Integration of Multiple Vehicle Models with an IMM ... - Semantic Scholar

2010 IEEE Intelligent Vehicles Symposium University of California, San Diego, CA, USA June 21-24, 2010

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Integration of Multiple Vehicle Models with an IMM Filter for Vehicle Localization Kichun Jo, Keonyup Chu, Kangyoon Lee and Myoungho Sunwoo, Member, IEEE Abstract— A vehicle localization system can be extremely useful for intelligent transformation systems (ITS) such as advanced driver assistance systems (ADASs), emergency vehicle notification systems, and collision avoidance systems. To optimize the performance of vehicle localization systems, localization algorithms that analyze multi-sensor data processed using a Kalman filter have been developed. However, a Kalman filter with a single process model cannot guarantee the accuracy of localization under various driving conditions, because the single vehicle model does not cover all driving situations. Therefore, we present a position estimation algorithm based on an interacting multiple model (IMM) filter that uses two kinds of vehicle models: a kinematic vehicle model and a dynamic vehicle model. While the kinematic vehicle model is suitable for low-speed and low-slip driving conditions, the dynamic vehicle model is more appropriate for high-speed and high-slip situations. The IMM filter integrates the estimates from a kinematic vehicle model based on an extended Kalman filter (EKF) and estimates from a dynamic vehicle model based on EKF to improve localization accuracy. The developed estimation algorithm was verified by simulation using a commercial vehicle model. The simulation results show that the estimates of vehicle position by the algorithm presented in this study are accurate under a wide range of driving conditions.

T

I. INTRODUCTION

HE localization system is an essential component of intelligent transportation systems (ITS). Several applications of ITS, such as advanced driver assistance systems (ADASs), electronic toll collection systems, emergency vehicle notification systems, and collision avoidance systems require highly accurate and reliable position information [1]. Nowadays, global positioning system (GPS)-based localization systems are widely used for these applications because GPS receivers allow the global position of a vehicle to be determined at low cost. However, positioning systems based on GPS alone can suffer from temporary loss of the satellite connection and signal error. To provide continuous, accurate, and high integrity position data, GPS-based localization systems should incorporate additional

Manuscript received May 15, 2010. Kichun Jo and Keounyup Chu are based in the Department of Automotive Engineering, Graduate School, Hanyang University, 17 Haengdang-dong, Seongdong-gu, Seoul 133-791, Korea ([email protected], [email protected]). Kangyoon Lee is a member of the Automotive Control & Electronics Laboratory, Hanyang University, Seoul 133-791, Korea ([email protected]). Sunwoo Myoungho is in the Department of Automotive Engineering, Hanyang University, 17 Haengdang-dong, Seongdong-gu, Seoul 133-791, Korea ([email protected]).

978-1-4244-7868-2/10/$26.00 ©2010 IEEE

sensors. The integration of GPS data with data from dead reckoning sensors such as inertial navigation systems (INS), encoders, and odometers by so-called fusion methods is common due to the complementary nature of these two types of sensors [2-7]. Even though a GPS provides absolute position information, the measurement frequency is low and discontinuous. Furthermore, signals from a GPS are affected by the external environment; GPS signals are blocked by tall buildings, tunnels, and dense foliage. In contrast, dead reckoning sensors give continuous measurements and are not interrupted by the external environment; however, they are prone to drift error due to accumulation of data. To integrate these two types of sensors, numerous Bayesian filter-based sensor fusion approaches have been proposed [1]. The extended Kalman filter (EKF) is widely used for information fusion algorithms because nonlinear localization problems can be easily solved with this filter [8-11]. The performance of a localization system depends not only on the filter structure, but also the appropriate choice of vehicle model. The role of vehicle models in localization systems was investigated in [12]. This paper demonstrated that a localization algorithm that has a good process model performed significantly better than an algorithm with a poor process model, although each algorithm had the same sensor suites and number of states. The major problem is that it is very difficult to choose the best model that covers all driving conditions. Higher order advanced vehicle models that can account for the various conditions are not suitable for localization systems because of the unobservability of some of the parameters of these models such as road conditions, tire properties, and vehicle type. As an alternative, the multiple model (MM) approach was proposed. This approach assumes that the system follows one of a finite number of different models. The possible vehicle driving patterns are represented by a set of models, and vehicle state information is obtained by combining specific model filters. Among several multiple model estimate methods, the interacting multiple model (IMM) estimator is the most popular due to its high performance and low computation power [13, 14]. Therefore, several previous studies have used the IMM filter for localization and tracking problems [15-17]. However, these studies only considered maneuvering and non-maneuvering driving conditions when designing the model set. In road vehicle applications, maneuvering conditions can be detected directly because

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steering input and wheel speed measurements are available. probabilities can be written as To cover various driving conditions, vehicle dynamic 1 (1) = µkj−|i1|k −1 = π ji µkj−1 i, j 1, 2. characteristic such as lateral force and tire slip angle should ci also be considered [18]. where µ kj−1 is the mode probability of the model j in the In this paper, we present a localization algorithm based on the IMM estimator that considers the dynamic characteristics previous step and π ji is the probability for the transition from of a vehicle. The vehicle model set comprises a kinematic vehicle model and a dynamic vehicle model. The kinematic model j to model i . The normalization coefficients are vehicle model is based on the assumptions of zero lateral 2 velocity and no wheel slip. This model is suitable for (2) ci ∑ = π ji µkj−1 i 1, 2. low-speed and low-slip driving conditions. However, in= the j =1 high-speed and large-slip driving region, the assumptions of The mixing initial state x ik −1|k −1 and the mixing initial the kinematic vehicle model break down, resulting in poor estimation performance. In contrast to the kinematic vehicle covariance P i k −1| k −1 of the model i are model, the dynamic vehicle model considers lateral tire force 2 and vehicle moments. Hence, it can estimate the vehicle (3) = = x ik −1|k −1 ∑ xˆ kj −1|k −1 µ kj−|i1|k −1 i 1, 2 lateral velocity and wheel slip, and can be applied to j =1 high-speed and large-slip driving conditions. By combining 2 T Pki−1| k −1 ∑ µ kj−|i1| k −1  Pˆk j−1| k −1 + ( xˆˆkj −1| k −1 − x ik −1| k −1 )( x kj −1| k −1 − x ik −1| k −1 )  = estimates from individual model-based EKFs, the IMM   (4) j =1 estimator can provide accurate localization information for a i = 1, 2 wide range of driving conditions. where xˆ kj −1|k −1 and Pˆk j−1|k −1 are the state and covariance of The rest of this paper is organized as follows. First, the overall system architecture is discussed in Section II. Next, the the model j of the previous step, respectively. system model set is defined in Section III. Finally, the simulation environment and results are described in Section B. Extended Kalman filter Using the mixing initial state and covariance of previous IV. process, the EKF of each model predicts and updates the model state and covariance. The overall operation of the EKF II. SYSTEM ARCHITECTURE for model i is shown in Fig. 2 [19]. At the prediction step, the Previous Interacting Extended Mode probability step Kalman Filter update predicted state xˆ i and its covariance Pˆ i are calculated. µkj−|i1|k −1 =

π ji µkj−1

2

∑π j =1

ji

µkj−1

µk1|1−1|k −1 xˆ 1k −1|k −1 Pˆ 1

µ

1|2 k −1|k −1

x 1k −1|k −1 P 1 k −1|k −1

k −1|k −1

µk2|2−1|k −1

xˆ 1k |k −1 Pˆ 1 k |k −1

xˆ 1k |k Pˆ 1

x 2k −1|k −1 P 2 k −1|k −1

xˆ 2k |k −1 Pˆ 2 k |k −1

k | k −1

Λ ik ci

2

∑Λ c j k

j =1

µ

k | k −1

At the update step, the EKF updates the corrected state xˆ ik |k

j

and covariance Pˆki|k using measurements.

1 k

k |k

Combination xˆ k |k Pˆk |k

Dynamic Model Extended Kalman Filter

µk2|1−1|k −1

xˆ 2k −1|k −1 Pˆk2−1|k −1

µki =

Kinematic Model Extended Kalman Filter

Prediction (1) Predict the state

xˆ ik|k −1 = fi (x ik −1|k −1 , uk ,0)

µk2

(2) Predict the covariance matrix

xˆ 1k |k Pˆ 1

δ fi δx

k |k

x = x ik −1|k −1

= Fki

T T Fig. 1. System architecture of the overall localization system. The system = Pˆki|k −1 Fki Pki−1|k −1 ( Fki ) + Gki Qk −1 ( Gki ) consists of an IMM filter that contains the kinematic vehicle model EKF and the dynamic vehicle model EKF.

The overall system architecture of the proposed localization system is shown in Figure 1. The system is composed of an interacting multiple model (IMM) filter that contains two extend Kalman filters: one for a kinematic vehicle model and the other for a dynamic vehicle model. The localization algorithm has four parts: an interacting, extended Kalman filter; a mode probability update, and components that combine the estimate values and covariance values. A. Interacting Initially, the states from the previous step of each model are mixed with mixing probabilities in the IMM filter. The mixing

Update (1) Compute the Kalman gain

K ki = Pˆki|k −1H T ( Ski ) −1

= Ski HPˆki|k −1H T + R (2) Update estimate with measurement i i i i xˆˆˆ k |k =xk |k −1 + K k ( zk − Hxk |k −1 )

(3) Update the error covariance

Pˆˆki|= ( I − K ki H ) Pki|k −1 k

Fig. 2. Extended Kalman filter for a kinematic vehicle model and a dynamic vehicle model.

C. Mode probability update The likelihood function of each mode i at time k , under the Gaussian assumption, is given by −1 T  1  exp  − (ν ki ) ( S ki ) ν ki  2  Λ ik =  i 2π S k

where ν ki and S ki are the innovation and its covariance. The model probability update is done as follows:

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(5)

1 i (6) Λ k ci c where c is the normalization constant for the mode probability update.

µki=

= c

2

∑Λ c j k

j =1

j

(7)

.

The kinematic vehicle model is reasonable for low-speed vehicle motion because the lateral force generated by the tires is very small. In contrast, at higher speeds or sharp steering, the no-wheel-slip assumption breaks down because the lateral force generates lateral velocities, and the vehicle slip becomes large enough to impact the system. In this situation, the dynamic vehicle model should be considered to assess the lateral velocity of the vehicle.

D. Estimation and covariance combination According to the Gaussian mixture equation, the combined state xˆ k |k and its covariance Pˆk |k are calculated as 2

xˆˆk |k = ∑ µki x ik |k

Y

O β δ −β

rF

(8)

rR

i =1

= Pˆˆk |k

2

∑µ i =1

i k

i i  P i + ( xˆˆˆˆ  k | k − x k | k )( x k | k − x k | k )   k |k  T

rG

VF

δ

VG

β

(9)

F

ψ

VR

G lf

R

III. VEHICLE MODEL SET The performance of the IMM-based localization algorithm depends on the design and selection of the model set. For superior performance, the model set should be able to represent a wide range of vehicle motion patterns. In this paper, the kinematic vehicle model and dynamic vehicle model are used to represent possible vehicle motions. A. Kinematic vehicle model The kinematic vehicle model was derived from the two degrees of freedom fundamental bicycle model shown in Fig. 3. In the bicycle model, the left and right front/rear wheels are represented by a single wheel. The front steering angle is described by δ . The center of gravity ( c.g. ) of the vehicle is

X

Fig. 3. Kinematic vehicle model. An assumptions of this model is that there is no slip between the wheels and the ground.

B. Dynamic vehicle model To take vehicle slip and lateral velocity into consideration, we developed a dynamic vehicle model for lateral vehicle motion. Equation (11) describes the lateral dynamics of the bicycle model. Eq. 11 was derived by summing the forces and moments about the center of gravity [20]. m and I z describe the vehicle mass and yaw moment of inertia, respectively, while Vx and Vy represent the vehicle longitudinal and lateral velocities, respectively. ∑ Fy = Fyf + Fyr = ma y = m (Vy + Vx r )

at point G . The distances from the c.g. to the front and rear wheels are represented by l f and lr , respectively. ( X , Y ) describes the orientation of the vehicle. The VG represents the velocity at the c.g. and it makes the slip angle β . The kinematic vehicle model makes the assumption that there is no slip between the wheels and the ground. Therefore, the steering angle is directly related to the yaw rate. Using the kinematics relationship, the following equation can be derived [20]:   VG cos (ψ + β )   VG sin (ψ + β )     VG cos ( β ) tan (δ ) / ( l f + lr ) 

where

 lr tan (δ )   l + l   f r 

β = tan −1 

(11)

∑ M z = l f Fyf − lr Fyr = I zψ

represent the location of c.g. in the global frame, and ψ

 X    = Y  ψ   

lr

The dynamic vehicle model makes the assumption that the lateral force that acts on a tire is proportional to the tire slip angle, as represented in (12). The tire slip angle can be defined as the angle of the wheel velocity vector relative to the longitudinal wheel axis. Y y x Vy

Fyr αr VR

(10)

G

Fyf VG

αf

VF

δ

β Vx ψ lf

lr

X Fig. 4. Dynamic vehicle model. An assumption of this model is that the lateral force that acts on a tire is proportional to the tire slip angle.

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TABLE I VEHICLE PARAMETERS

l ψ   Fyf= 2C f α f ≈ 2C f  β + f − δ  Vx    lrψ  Fyr =2Crα r ≈ 2Cr  β −  Vx  

(12)

Vehicle parameter Mass

C f and Cr represent the front and rear tire cornering

m

Yaw moment of inertia

Iz

Value

Unit

1653

kg

2765

kgm 2

Distance from the

stiffness, respectively. The tire slip angle of the front and rear wheels are represented by α f and α r , respectively.

c.g. to the front wheel l f

1402

mm

Distance from the

c.g. to the rear wheel lr

1646

mm

We calculated the overall vehicle motion of the dynamic vehicle model using the following equation:

Front tire cornering stiffness

Cf

183350

N / rad

Rear tire cornering stiffness

Cr

154700

N / rad

         

TABLE II SENSOR PROPERTIES

(13)

The proposed IMM-based localization system integrates the EKF estimates of each model using the mode probability to adapt to a wide range of driving conditions. Under high-speed and large-slip driving conditions, the dynamic vehicle model-based filter is more appropriate because it considers lateral velocity and the slip of a vehicle. In contrast, under low-speed and small-slip driving conditions, the kinematic vehicle model-based filter provides better localization estimates because the process noise of the lateral velocity of the vehicle does not affect the filter estimation process. IV. SIMULATION The proposed localization algorithm was analyzed by intensive simulation studies. The simulation results presented in this paper are averages of 100 Monte Carlo simulations. The reference models for the vehicle and sensors were designed using the commercial vehicle dynamic simulator, CarSim. Because Carsim has 27 degrees-of-freedom math models, it can provide extremely accurate vehicle dynamic information. Furthermore, vehicle parameters can be obtained using this simulator, as shown in Table I. A. Measurement system The simulation vehicle was assumed to be equipped with in-vehicle sensors and a GPS receiver. Nowadays, the vehicle chassis control system is an essential component of the vehicle safety system. This system requires sensor equipment such as a yaw rate sensor, wheel speed sensors, and a steering angle sensor. These measurements are available from in-vehicle sensor networks, such as the controller area network (CAN), without additional cost. Table II describes the properties of the equipment sensors. The noise from each sensor was assumed to be Gaussian white noise. Position information from a GPS receiver was emulated by summing the position

Sensor

Noise (1 σ )

Unit

Yaw rate sensor

0.05

deg/ s

Steering angle sensor

0.1

deg

Wheel speed sensor

0.1

GPS position

10

m/s m

from the true vehicle model and the assumed GPS noise. The in-vehicle sensors were emulated similarly. The wheel speed sensor information and steering angle sensor information were used as inputs for the process model during the prediction step of EKF. The yaw rate sensor and GPS receiver were used to measure the update step. B. Steady-state cornering test Vehicle lateral dynamic characteristics were analyzed using a steady-state cornering test. The test was performed with a constant steering angle and two kinds of constant speed. The kinematic vehicle model EKF and the dynamic vehicle model EKF were compared. To analyze the low-slip dynamics of the vehicle, a constant steering angle input of tow degrees and a constant velocity (speed of 20 km/h) were maintained for 120 seconds. The results of the low-speed steady-steering test are shown in Fig. 5. Because the lateral slip was very small, the kinematic 2.5 Kinematic EKF Dynamic EKF 2

Distance Error [m]

Vx cos(ψ ) − Vy sin(ψ )   V  y cos(ψ ) + Vx sin(ψ ) X        2C l 2 + 2C l 2  2C f l f  2 C f l f − 2 C r lr  f f r r  Y  = −  ψ −  Vy + ψ    I zVx I z Vx Iz        2 C f l f − 2 C r lr  2C f δ  2 C f + 2 Cr  Vy    ψ −  V y +  −  Vx + Vx m m   Vx m   

1.5

1

0.5

0

0

20

40

60 Time [sec]

80

100

120

Fig. 5. Steady-state cornering test under low-slip conditions. A steering input of 2 degrees and a constant velocity of 20 km/h were maintained for 120 seconds.

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5 Kinematic EKF Dynamic EKF

4.5 4

Distance Error [m]

3.5 3 2.5 2 1.5 1 0.5 0

0

5

10

15

20 Time [sec]

25

30

35

40

Fig. 6. Steady-state cornering test under high-slip conditions. A steering input of 2 degrees and a constant velocity of 100 km/h were maintained for 40 seconds.

sensor input from the simulation vehicle following this road profile. 2) Simulation results: The tire slip angles obtained from the true model are shown in Fig. 9, and the mode probability for each model is described in Fig. 10. In the small slip cornering region, the mode probability of the kinematic vehicle model was larger than the mode probability of the dynamic vehicle model. Based on this result, we can surmise that the kinematic vehicle model is more suitable for low-slip driving conditions than the dynamic vehicle model. In contrast, in the large-slip cornering region, the mode probability of the dynamic vehicle model was larger than the mode probability of the kinematic vehicle model, indicating that the dynamic vehicle model is more appropriate than the kinematic vehicle model for high-slip regions. 40 velocity [m/s]

vehicle model was used to represent the vehicle motion. The estimate of the kinematic vehicle model EKF had less error than the dynamic vehicle model EKF estimate. Under high-slip driving conditions, a constant steering angle input of 2 degrees and a constant velocity of 100 km/h were maintained for 40 seconds. The results of the high-speed steady-steering test are shown in Fig. 6. The dynamic vehicle model EKF showed better performance than the kinematic vehicle model EKF in the IMM-based localization system simulation because the dynamic vehicle model is more suitable for high-slip driving conditions than the kinematic vehicle model EKF.

Y [m]

30

20

40 Time [s]

50

60

70

80

60

70

80

60

70

80

Gyro sensor [deg/s]

10 0 -10

0

10

30

20

40 Time [s]

50

40 Time [s]

50

(b) Yaw rate sensor Steering Angle [deg]

15 10 5 0 -5

0

10

20

30

1

50

0

Start Point

0

Slip angle [deg]

-1

-50 -250

velocity [km/h]

10

(a) Wheel speed sensor

100

-200

-150

-100

-50

0 X [m]

50

100

150

200

-2

-3

100

Left-front tire Left-rear tire Right-front tire Right-rear tire

-4

50 0

0

(c) Steering angle sensor Fig. 8. Sensor measurements. (a) Wheel speed measurement (b) Yaw rate measurement (c) Steering angle measurement.

40 sec

150

10

20

250

50 sec

20

0

C. Evaluation of several slip conditions 1) Driving scenario: The road map for the simulated localization algorithm is shown in Fig. 7. A driver model that was provided by CarSim followed this road map with the velocity profile shown in Fig. 7. Figure 8 shows the emulated

200

30

0

10

20

30

40 Time [s]

50

60

70

-5

80

Fig. 7. Driving map and velocity profile for simulating the proposed IMM-based localization algorithm.

0

10

20

Fig. 9. Each tire slip angles.

750

30

40 Time [sec]

50

60

70

80

algorithm is accurate and reliable enough under various driving conditions for vehicle implementation.

0.9 Kinematic Model Dynamic Model

0.8

ACKNOWLEDGMENTS

Mode Probability

0.7

This study was financially supported by the Ministry of Knowledge Economy (MKE) and the Korea Institute for Advancement in Technology (KIAT) through the Workforce Development Program in Strategic Technology, and was partially supported by the Brain Korea 21 Project in 2009.

0.6 0.5 0.4 0.3

REFERENCES

0.2 0.1

0

10

20

30

40 Time [sec]

50

40 Time [sec]

50

60

70

80

Fig. 10. Mode probabilities in the IMM localization algorithm. 14 Kinematic EKF Dynamic EKF IMM

12

Distance Error [m]

10 8 6 4 2 0 0

10

20

30

60

70

80

Fig. 11. Comparison of distance errors.

Fig. 11 describes the distance errors from the true position for each filter. In the small-slip region, the EKF of the kinematic model provides better performance than the dynamic model. In contrast, for the high-slip region, the EKF of the dynamic model has a smaller error than the dynamic model. The proposed IMM-based localization system combines these two EKF estimates with the mode probability shown in Fig. 10, and thereby offers more precise estimates of position under various driving conditions. V. CONCLUSIONS In this paper, we described the development of a vehicle localization system based on information from a GPS receiver and in-vehicle sensors. The proposed IMM-based localization system combines the EKF estimates from a kinematic vehicle model and a dynamic vehicle model to adapt to several driving conditions. Under high-slip driving conditions, the dynamic vehicle model-based filter is used to take lateral velocity and vehicle slip into consideration. In contrast, for the low-speed and small-slip region, the kinematic vehicle model-based filter is used because the process noise of the lateral velocity of the vehicle does not affect the filter estimation process. We verified this localization algorithm by extensive simulations using a commercial vehicle model. The simulation results showed that the estimation accuracy of the developed

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