Proceedings of the 2008 IEEE International Conference on Information and Automation June 20 -23, 2008, Zhangjiajie, China

Research on Real-time Collision Detection for Vehicle Driving in the Virtual Environment* ˈ2*

Guan Jianhong1 1

, He Hanwu1, Zhai Wenxuan1, Liang Yanfei1

Faculty of Electromechanical Engineering, Guangdong University of Technology, No.729 Dongfengdong Road, Yuexiu District, Guangzhou, China 2 School of Mechanical Engineering, South China University of Technology, No.381 Wushan Road, Tianhe District, Guangzhou, China *Email: [email protected]

Abstract – Collision detection plays a critical role in the immersion and authenticity of virtual driving as a virtual reality system. Through analyzing and comparing several common algorithms of collision detection, a method of real-time collision detection was further developed, including two detecting phases, pre-collision detection and accurate-collision detection. First of all, Radar ranging theory was used in the pre-collision detection as it could predict the possible colliding region to save computer resources effectively. If the pre-collision were true, one collision type would be chosen which determined by the relative direction of two vehicles. Subsequently, an accurate collision detection were about to engage in corresponding to the collision type. Every collision type has its own collision detection algorithms. A mathematic model was built up for the minimal collision distance which ensured the real-time and authenticity of vehicle driving in virtual environment. The results showed that the method was simple and met the demand of real-time, which satisfied the detecting requirements of vehicle driving in the virtual environment. Moreover, it can also be used in other collision detection experiments. Index Terms – Real-time Collision Detection, Vehicle Driving, Virtual Environment, Radar Ranging.



1. INTRODUCTION*

Collision detection is very important to get immersive performances for 3-Dimensional objects in the virtual reality system. Research on collision detection is quite a necessary for virtual driving acting as a virtual reality system [1]. The algorithms of collision detection are different according to different applications. For example, for the vehicle driving in virtual environment, it concentrated on the collision between vehicle and vehicle or between vehicle and other objects in the virtual environment. According to statistics, vehicle impacts occupy nearly up to 90% among all reasons of road traffic accidents [2]. It’s not difficult to detect collision which is to calculate interaction between two polyhedrons mathematically. However, in a real-time simulation, it calls for high time requirement, in other words, collision detection *

This research was supported by National Natural Science Foundation of China (50775047), Guangdong provincial science and technology project (2005B10201021) and Natural science foundation of Guangdong province (06021445).

978-1-4244-2184-8/08/$25.00 © 2008 IEEE.

algorithms are required to realize the real-time performance. An algorithm of the real-time collision detection algorithms was developed through comparing several common methods according to the simulation requirements of vehicle driving. Rapid collision detection plays a crucial role in improving the authenticity and the immersion of virtual environment; however, higher requirements are put forwards for collision detection because of the complexity, real-time performance and accuracy of it. Ref. [1] developed a method of collision detection for points and convex polygon in freeway virtual environment. Ref. [3] presented a simplified mathematical model for vehicle collision detection. It was to estimate the distance of two Cg (Shorted for center of gravity) points and maximal and minimal detecting radius. A simple and inexpensive mathematical model was established for the collision detection of vehicle driving in [4]. A new k-Dops hierarchical bounding box model of collision detection [5] was presented after theoretical analysis. In Ref. [6], laser was used as vehicle radar to measure the distance between itself and the obstacles in front of the vehicle by researching different laser emitter and semiconductor laser emitter was used in the rear-end collision avoidance system. In the 6th 3D.VR Simulation Contest, the system of Meijo University was currently under development, as in [7]. It can evaluate driving abilities of senior citizens by integrating the relevant features of the program like range of view. University of Parma [8] introduced a vehicle detection algorithm based on a generally symmetric vehicle. It was characterized by a rectangular bounding box which satisfied specific aspect ratio constraints. In order to get a real driving environment and to make vehicle driving safe, as in [9], Toyota developed a World-class Driving Simulator which was real-as-possible environment to aid development of active safety technology; Mercedes applied ranging radar to vehicle safety research and used radar sensors to analyze the size and weight of a potential accident partner. 2. BASIC CONCEPTS O F C OLLISION D ETECTION In brief, collision detection is to detect whether different objects collide with each other at a second in the virtual environment [10]. Taking time into account, collision detection can be divided into continuous collision detection

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and discrete collision detection. Continuous collision detection can be defined as below: Definition: Suppose a three-dimensional space R is denoted by three-dimensional coordinate system Fw . There are N motion models whose positions and attitudes are varying over time. Fi specifies the space occupied by No.i. model. A four-dimensional coordinate system Cw is formed when Fw is changing over time. The four-dimensional coordinate system Ci comes into being if model Fi varied along a certain track of motion. Collision detection is to judge whether formula C1  C2  C3 ...  Cn = Φ comes in existence or not. Collision detection algorithms can be expressed in a common flow (See Fig.1). 3. G ENERAL A LGORITHMS O F C OLLISION D ETECTION 3.1. Space Decomposition Method Space decomposition is to break down the virtual space into many equal volume grids and tests objects inhibited in the same grid or sideward grids, as in [11]. Typical methods include Octree method and Binary Space Partitioning (BSP). The principle to the Space Decomposition is that the two objects can not overlap if they are in different grids and further overlapping test is needed if two objects locate in the same grid. 3.2. Bounding Volume Hierarchical Bounding Volume Hierarchical is to wrap the complicated geometrical objects with big volume and simple shape boxes. When detecting collision, we should do overlapping tests among bounding boxes at first. Then geometrical objects should be detected accurately [12, 13]. Obviously, bounding volume method is effective to judge whether two geometrical objects are overlapped. There are many types of bounding volumes: Bounding Spheres, Axis-aligned Bounding Boxes (AABB), Oriented Bounding Boxes (OBB), discrete orientation polyhedrons (k-DOP), and many more. Their algorithms are as following: 1) Bounding Spheres method The bounding sphere can be defined as the minimal sphere which can wrap the object. Ref. [14] specifies their region: R = {( x, y, z )T | ( x − cx ) 2 + ( y − c y )2 + ( z − cz ) 2 < r 2 } (1) for t: from 0 to t-end, step is Δt for object Ai ∈ (A1, A2, …, An) move Ai to the position at the time of t for object Aj ∈ (Ai+1, Ai+2, …, Ai+n) move Aj to the position at the time of t for every polygonal Pi in object Ai for every polygonal Pj in object Aj if (Pi interacts with Pj) then Ai collides with Aj at the time of t

A reasonable representation for a sphere is the center point c (cx , c y , cz ) and radius r . It requires four parameters. 2) Axis-aligned Bounding Boxes (AABB) method The axis-aligned bounding box (AABB) [15] was defined as a minimal cube wrapped the object and its edges paralleled the coordinate axis. Therefore, to describe an AABB only needs 6 scalar quantities. When constructing AABB, its direction should be the same with the axis direction of local coordinate. So all AABB have the same direction. They specify the region: R = {( x, y, z )T | lx ≤ x ≤ ux , l y ≤ y ≤ u y , lz ≤ z ≤ u z } (2) Where l x , u x , l y , u y , l z and u z are the min and max value along each axis. 3) Oriented Bounding Boxes (OBB) method The OBB was defined as an arbitrary cube which wrapped the object and had a direction relative to the coordinate axis. Compared with AABB, the unique trait of OBB is its arbitrary direction, which can wrap the object tightly according to the shape [16]. It specifies the region: R = {C + ar1v1 + br2 v2 + cr3 v3 | a, b, c ∈ (−1,1)} (3) Where, the central point is C ; v1 , v2 , v3 are the three vertical directions and r1 , r2 , r3 are radiuses on three directions. 4) k-Discrete Orientation Polyhedrons (k-DOP) method The principle of k-DOP is to approach object with more than 3 pairs of parallel planes which are the intersection of two half spaces fixed by two changeless normal vectors with opposite directions [17]. As to the practical application, we should choose the direction of normal vectors effectively to predigest the parallel pair-wise planes.

3.3. Comparison Conclusions (1) For Space decomposition method, when the objects distribute uniformly and in a far distance, the calculating efficiency will be high for collision detection. If there are too many objects in the space or they locate very close, the efficiency will be very slow. (2) For the Bounding Volume Hierarchical method, although it can prevent the phenomenon of “penetration” when objects moving very quickly, the moving speed should be limited and the motion track should be simple. This method will limit the application ranges. Moreover, it’s quite difficult to set up models for this method, especially for complicated and sudden motions. (3) More complex the bounding boxes would be, tighter the virtual objects wrapped. It’s a pair of contradiction. So the key problem is how to keep balance between its simpleness and tightness at the same time. (4) There is no algorithm which can be applied to all cases because most of collision detection algorithms are designed for idiographic applications. Furthermore, there are few high-effective detection

Fig.1 Common flow of collision detection algorithms

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algorithms for the dynamic and complex virtual environment. After synthesizing and comparing those common detecting methods, this paper presented a real-time collision detection algorithm of vehicle driving in the virtual environment.

Area of precollision detection

4. P RINCIPLE O F REAL-T IME C OLLISION D ETECTION

Car

4.1. Vehicle driving characters In order to make the colliding effect real and real-time for vehicle driving in the virtual environment, our vehicle driving has following features. (1) In the driving environment, there are many vehicles and objects staying along the roads to be detected. More than 100 objects need to be tracked dynamically. (2) It not only needs to detect the collisions between moving objects and static ones, but also needs to detect collisions between two moving objects. (3) It requires to judge whether the distance between two cars is equal or less than the minimal collision distance. (4) According to the classification of vehicle impacts, this paper mainly studies the three impacts, such as frontal impact, rear-ends collision and side impact. 4.2. The principle The collision detection includes two phases, the first one is the phase of pre-collision detection and the second one is the phase of accurate collision detection. In the first phase, theory of radar ranging was used to determine the capable collision region rapidly and then further operations would be taken for these potential collision regions [18]. The program had to calculate each point for every step Δt , which may occupy large system resource and also can not detect in real time. In the second phase, different detecting algorithms can be chosen in terms of different collision types. The frame location was shown in Fig.2. 1) Phase of pre-collision detection When driving in virtual environment, the vehicles were impossible to response after colliding with others. It should take actions when the distance was up to a lower distance value. As for this purpose, the radar ranging method was used to predict a potential area of collision detection [19] (See Fig.3). Phase of Pre-collision Detection

Phase of Accurate Collision Detection

Fig.2 Fra me of collision detection

Fig.3 Pre-Collision Detection area

When running, a round pre-collision detection area would be formed by circling around the Cg (shorted for Centre of Gravity) point and rotating at a certain speed. Its radius is the radius of pre-collision detection. If Dis _ P , which specified the distance between two vehicle Cg points, a R satisfied the following formula, the system would treat with the pre-collision between two vehicles. Dis _ P − l2 ≤ R (4) In this formula, l2 was expressed as the distance from Cg Point to the vehicle’s most frontal point. 2) Phase of Accurate Collision Detection The accurate collision detection would be started after pre-collision. In this phase, at first, the possible collision type was determined according to the vehicles’ relative direction angle which was expressed as α . If α was 180 degree, the detection program of frontal impact would be started; if α was 0 degree, the detection program of rear-end collision would be started; and otherwise, the detection program of side impact would be started. The flow chart of the detecting process was shown in Fig.5. This experiment was conducted on the below assumptions: (1) If side impact was queried, rectangle bounding boxes would be used to simplify the vehicle models. (2) There were only three freedoms of motion: two translations ( X and Y displacements) and rotational motion (around Z axis). (3) Vehicle and road were on the same plane before or after collision. (4) Minimal collision distance μ was used to ensure that the system had adequate time to calculate and response in time. 5. MATHEMATIC MODEL OF MINIMAL COLLISION DISTANCE Minimal collision distance μ is also called as the vehicle’s safety distance which has great relative to vehicle braking distance. So the braking distance should be calculated before calculating the minimal collision distance. There are four phases for vehicle braking process [18, 20, 21, 22]: driver response time t1 , brake coordinate time t 2 , deceleration increasing time t3 and continual brake time t 4 . Relation between braking deceleration and time and relation between braking distance and time were simplified as shown in Fig.4.

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t1

t2

t3

t4

Time t

Brake Distance S

Brake Deceleration a

S3 = V0t3 3.6 − 1 6 amax t32

Time t

Fig.4 Simplified relation between braking deceleration a and time t and relation between braking distance S and time t

1) Calculating this vehicle’s braking distance The initial speed and maximum deceleration of this vehicle were supposed as Vo and amax , their units are km/h and m/s2 respectively. (1) Running distance S1 during Driver response time t1 Because there was no brake force during the period of driver response time t1 , vehicle still run on the braking initial velocity V0 and its running distance S1 can be calculated by below formula: S1 = V0t1 3.6 (5) (2) Running distance S 2 during brake coordinated time t 2 Because there was no brake force during the period of brake coordinated time t 2 , vehicle still run on the braking initial velocity V0 and its running distance S 2 can be calculated by below formula: S2 = V0t2 3.6 (6) Brake coordinated time t 2 is made up of the delay time of braking transmission and increasing time of brake force. Its value was not only fluctuated by pedaling speed, but also influenced by the structure of arrester. For the hydraulic arresters, t 2 was from 0.25 to 0.95; for pneumatic arresters, t2 was from 0.35 to 0.95, as in [23ˈ24]. (3) Running distance S3 during deceleration increasing time t3 In this period, the deceleration was increasing linearly, so it can be assumed as below formula. dV dt3 = kt3 (7) V

t3

V0

0

Where: k = − amax t3 , So: ³ dV = ³ kt3 dt3

(8)

Due to the initial velocity of this period was: V30 = V0 Ÿ V = V0 + 1 2 kt32 Ÿ dS dt3 = V0 + 1 2 kt

2 3

(9) (10)

Ÿ ³ dS = ³ (V0 + 1 2 kt )dt3 2 3

(11)

At initial time: S = 0 Ÿ S3 = V0t3 + 1 6 kt Ÿ S3 = V0 t3 − 1 6 a t Change the unit of velocity into km/h, so: 2 max 3

3 3

(12) (13)

(14) According to the relevant statistical analysis, t3 can be considered as 0.1s in actual braking process, and vehicle deceleration is less than car 10m/s2 commonly [23]. The end part of the above formulate can be ignored and it can be simplified as below. S3 = V0t3 3.6 (15) (4) Running distance S 4 during continual braking time t 4 During this period, vehicle run on the even deceleration amax , its initial velocity is Vo and the end-velocity was 0. Referring to corresponding formulates S 4 could be calculated by following formula. S 4 = V0 2 (2 × 3.62 amax ) (16) Combined above formula, the braking distance can be calculated. S = S1 + S2 + S3 + S4 (17) = V0 (t1 + t2 + t3 ) 3.6 + V02 (2 × 3.62 amax ) 2) Braking distance of objective vehicle The braking process of objective vehicle was similar to this vehicle’s. This vehicle started to be aware of other vehicles which were nearby and might cause colliding until the objective vehicle decelerated. Therefore, braking distance of objective vehicle can be formulated as below. Si = V i2 (2 × 3.62 ai max ) (18) Where: the initial speed and braking deceleration of the objective vehicle are specified as Vi and ai max . Therefore, vehicle’s safety distance can be calculated by below formula: S ' = S − Si = V0 (t1 + t 2 + t3 ) 3.6 + V0 2 (2 × 3.62 amax ) − Vi 2 (2 × 3.62 ai max ) (19) The warning condition of collision is: (20) D≤S' Where: running distance between two vehicles is expressed as D . The time would be delayed during the course of calculation and judgment in the application. Therefore, redundancy value ε should be added when calculating the minimal collision distance μ whose mathematic model was presented as follow: μ = S '+ ε (21)

6. F LOW C HART O F P ROGRAM A ND D ATA S TRUCTURE Firstly, query the current Cg Coordinates of two vehicles and calculate the distance (shorted by Dis) between Cg P1 and P2; then verdict the relation between Dis and radius of pre-collision detection. If pre-collision was true, accurate collision detection would be started and the entire flow chart was shown in Fig.5. Data structures which were involved in vehicle collision detection included: int CollissionType; // Collision type which included frontal impact, rear-end collision and side impact boolean pre_CollissionFlag˗ // Pre-colliding sign

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boolean CollissionFlag˗ // colliding sign float* P_Coordinate; // Coordinate Cg of vehicles float* Vertex_Coordinate; // Vertex Coordinates of two rectangle bounding boxes for vehicles. float MinCollisionDistance˗// Minimal collision distance float CollisionDetectRadius˗// Collision detection radius CheckCollision (Vehicle_Movement*); // Function of collision detection, which checked collisions every Δt Vehicle_Movement { 3DModel* model; // Point to a 3D Model Class, which can store the information of 3D Model. Vector Pos˗// Coordinates of vehicle in the virtual scene Vector V˗// Vehicle velocity }

completely by only one coordinate axis. One-dimensional impact is also divided into frontal impact and rear-end collision. In the frontal impact, two vehicles are on the opposite directions before colliding. Reversely, as to a rearend collision two vehicles are on the same directions before colliding because rear vehicle was faster than the front one and the head of rear one touched the back of the frontal one. The coordinate system of two rectangular and one rotate angle are used to describe the vehicle positions before or after colliding in the two-dimensional impact. Fig.6, Fig.7 and Fig.8 show the sketch maps of frontal impact detection, rear-end collision detection and side impact detection, as well their forms in the virtual environment respectively at a certain moment.

V2

R

V1

Typedef Struct { float x; float y; float z; }Vector;

a

P1

Car A

P2

a1

Car B

a2

D is_P

Start

Fig.6 Sketch map of frontal impact detection and its form in virtual environment.

Query the current Cg Coordinates of two vehicles

No

Calculate the distance Dis between Cg P1 and P2; Calculate the pre-collision detection radius R; R

L2

V1

Dis-a2=R

V2 a

P2

P1

Yes

Car B

Car A

Two vehicle collide, pre_CollissionFlag=True

a2

a1

Get the relative direction angle of two vehicles Dis_P

Query the collision type

Fig.7 Sketch map of rear-end collision detection and its form in virtual environment

1D Impact 2D Impact Rear-End Collision

a2

Query two vehicles’ No Position parameters and No state parameters; Calculate Calculate vertex the Distance Dis_P Coordinates of two between two Cg points. rectangle bounding boxes.

abs(Dis_P-a1-a2)<μ

Di

s_P

B2

P2

B4

V2 B1

Car B

V1

A2 Car A A3

i=1 to 4; j=1 to 4

a

No Query two vehicles’ Position parameters and state parameters; Calculate the Distance Dis_P between two Cg points.

B3

Query two vehicles’ Position parameters and state parameters;

R

Frontal Impact

A1 P1

A4

abs(Dis_P-a1-(L2-a2))<μ

Fig.8 Sketch map of side impact detection and its form in virtual environment.

Judge whether the vehicle Vertex touches the side of another vehicle.

Yes

CONCLUSION Yes

Yes Collide with each other, CollissionFlag=True End

Fig.5 Flow chart of collision detection program

Vehicle Crash type can be divided into one-dimensional impact and two-dimensional impact according to the direction of velocity [25]. In the one-dimensional impact, the motion of two vehicles’ Cgs keep on the same line all the time and two vehicles’ motion state can be described

In this paper, we present a real-time algorithm for collision detection to enhance the real performance in the virtual driving. Pre-collision detection and accurate collision detection were used for this algorithm. In the pre-collision detection, the radar ranging method prevented unnecessary detections and decreased the system load. The accurate collision detection can detect the three different impacts and make virtual driving more genuine. The collision detecting program of vehicle driving was developed under the virtual reality software EON and Visual C++ 6.0. We also set up the

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mathematic model of minimal collision distance which would be meaningful for other virtual reality system in use for reference. REFERENCES [1] Wang Huasen, Xiong Jian, and Yi Tao, “Application of the method for collision detection in the virtual simulation of vehicle,” J. Journal of Kunming University of Science and Technology (Science and Technology), vol. 29, no. 2, pp. 136-138, April 2004, (In Chinese) [2] Wang Jie, “Analysis on the ranging performance of automotive anticollision radar,” J. Sci-tech Information Development & Economy, vol. 17, no.19, pp. 198-199, 2007, (In Chinese) [3] Huang Jingan, and Shen Peimin, “The application of collision detection to traffic accident simulation,” J. Computer and Communications, vol. 22, no.1, pp. 47-48, 2004, (In Chinese) [4] Zeng Junwu, Guo Qisheng, and Li Bing, “A simplified mathematical model for collision detection in vehicle driving,” J. Computer Simulation, vol. 17, no.6, pp. 13-15, Nov 2000, (In Chinese). [5] Tu Weimin, “Researches on collision-detection in virtual-surgery,” D. Heilongjiang: HarBin Engineering University, M.S. Degree, 2007, (In Chinese) [6] Deng Mingzhe, “Research for rear-end collision alarm system on high way,” D. Hubei: Wuhan University of Technology M.S. Degree, 2006, (In Chinese) [7] Meijo University, “Driving Ability Measurement for Senior Citizen VR Simulation,” http://www.forum8.co.jp/english/uc-win/VRcontest-review-6e.htm, 6th 3D.VR Simulation Contest, UC-win Road [8] “Vehicle Detection,” http://www.argo.ce.unipr.it/argo/english/index.html , University of Parma, Italian. [9] “Mercedes Looks to Safety Future,” http://qiche.info/auto168/bencandy .php?fid=106&id=1408, Daimler Chrysler AG, Eds. Frank de Leeuw van Weenen, June 2007 [10] Wang ZhaoQi, Zhao QinPing, and Wang ChengWei, “An objectoriented collision detection method and its application on distributed virtual environment,” J. Chinese Journal of Computers, vol. 21, no. 11, pp. 990994, Nov 1998, (In Chinese) [11] Huang Juan, and Gu Jinan, “Summary of the research in collision and interference detection in assembly simulation,” J. Journal of Jiangsu University of Science and Technology, vol. 23, no. 2, pp. 17-21, March 2002, (In Chinese)

[12] Wei Yingmei, et al, “Research on fixed direction hull bounding volume in collision detection,” J. Journal of Software, vol. 12, no. 7, pp. 1056-1063, 2001, (In Chinese) [13] A. Garcia-Alonso, N. Serrano, and J. Flaquer, “Solving the Collision Detection Problem,” J. IEEE Computer Graphics and Applications, vol. 14, no.3, pp. 36-43, May 1994 [14] Senfan Gottschalk, “Collision queries using oriented bounding boxes,” D. University of North Carolina, USA, 2000 [15] Wang Xiaorong, “Research on collision detection algorithm based on aabb bounding volume,” D. Shanghai: HuaZhong Normal University, M.S.Degree, May 2007, (In Chinese) [16] P.M.Hubbard, “Collision detection for intersection graphics application,” J. IEEE Transactions on Visualization and Computer Graphics, vol. 1, no. 3, pp. 218-230, Sep 1995 [17] J.T Klosowski, et al, “Efficient collision detection using bounding volume hierarchies of K – DOPs,” J. IEEE Transactions on Visualization and Computer Graphics, vol. 4, no. 1, pp. 21-36, Jan-Mar 1998 [18] Zhang Dabiao, and Wang Yanju, “Design of microwave velocity and distance monitor system,” J. Instrument Technique and Sensor, vol. 4, no. 4, pp. 11-13, 2004, (In Chinese) [19] Lu Yongming, “Virtual traffic environment modeling towards experimental platform for driving safety control,” D. Guangzhou: Guangdong University of Technology, M.S. Degree, May 2007. (In Chinese) [20] Dang Hongshe, Han Chongzhao, and Duan Zhansheng, “Determination of distance for vehicle collision avoidance warning and braking,” J. Journal of Chang’ an University (Natural Science Edition), vol. 22, no. 6, pp. 89-91, 2002, (In Chinese) [21] Chen Guangwu, Hou Dezao, Li Xiaoxia, and Li Baichuan, “Calculating model of practice and safe space between car heads on express way,” J. Chinese Ergonomics, vol. 7, no. 1, pp. 41-44, March 2001, (In Chinese) [22] Xiang Qiaojun, Wang Wei, and Li Wei, “A study on the vehicle minimum time headway,” J. Journal of Southeast University (Natural Science Edition), vol. 28, no.3, pp. 79-82, May 1998, (In Chinese) [23] Yu Zhisheng, Automobile Theory, Second edition, Beijing, China: China Machine Press, 1990. (In Chinese) [24] [West Germany] Manfred Mitschke, and Chen Yinsan, Translated. Automobile Dynamics, Beijing, China: China Communications Press, 1997. (In Chinese) [25] Li jiang. Traffic accident Dynamics [M]. Beijing, China: China Machine Press, Feb 2000, (In Chinese)

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IEEE Paper Template in A4 (V1)
Security. INTRODUCTION. Peer-to-peer (P2P) or ad-hoc networking has become one of the ... scheme is to control peers lying about their services they provide.

Additional illustrations related to the IEEE SPL paper ...
Additional illustrations related to the IEEE SPL paper, “Testing the Energy of Ran- dom Signals in a Known Subspace: an Optimal Invariant Approach”.