Agent Based Intersection Traffic Simulation

Transcription

1 Agent Based Intersection Traffic Simulation David Wilkie May 7, 2009 Abstract This project focuses on simulating the traffic at an intersection using agent-based planning and behavioral methods. The motivation is to simulate this traffic in a way that s noisy, believable, and high level of detail. The main approach is based on the idea of a control obstacle and a greedy sampling method to navigate the robots. Other behaviors are generated using constraints and finite state machines. 1 Introduction Existing traffic simulators focus on simulationg large traffic networks. They do so at the expense of realistic graphics. Each car moves along a track and thus doesn t have the noisy, real-workld feel of traffic. This approach is good for modeling and testing road designs, but isn t suitable for high level of detail animation. Traffic is a part of everyday life, and it thus shows up in movies and television shows. An agent based approach is also ideal for video games, as in Figure 1, in which the cars may need to react believable to the unanticipated actions of a player. Agent based simulations of traffic are also closely connected to robotics, and thus there are also applications in automated transportation systems for factories, as shown in Figure2, and other systems. We want the agents to move according to the kinematic constraints of a car. Additionally, they should behave human-like.they should stay in lanes, wait for a green light, avoid collisions, and have imperfect driving. The approach presented in this paper uses greedy Figure 1: Traffic in a modern video game. sampling in control space to find controls for each car. Additional constraints and considerations are added to get desired effects. The approach is based on the idea of control obstacles, which is a generalization of velocity obstacles.[2, 1] 2 Prior Work 2.1 Non-holonomic Planning Numerous approaches exist to planning for nonholonomic robots. One of the early works, [6] proved that a path for a holonomic robot could be transformed into a feasible path for a non-holonomic robot under the condition that the holonomic path lay within an open region of the free configuration space, and [6] also provided an algorithm for making this transformation. Smooth planning for non-holonomic robots was approached in [9]. 1

2 Over 100 models total from engineering, mathematics, operations research, physics, and computer science Microscopic Models Microscopic models assume the acceleration of car α is dependent on neighboring vehicles. The primary influence is the leading vehicle, α 1, i.e. the car ahead. The model of behavior is then Figure 2: Robots moving shelves in a warehouse. 2.2 Multiple Robots with Nonholonomic Constraints Numerous approaches also exist for complete planning for multiple robots with non-holonomic constraints [4, 3, 11]. However, a problem with complete planning of this sort is he high dimensionality of the spaces planned within. One method of getting around this problem is to decompose the problems of path generation from velocity planning. This is the approach of [5, 8, 10], in which paths are first found, then velocity profiles for each robot are searched for to avoid collisions. 2.3 Traffic Models Many simulation models of traffic have been created over the years, with particularly strong progress being made in the last decade. A brief timeline of when various approaches were developed follows Timeline of Modeling Approach Invention 1950s Microscopic (follow-the-leader) models 1950s Macroscopic (fluid-dynamic) models 1960s Mesoscopic (gas-kinetic) models 1990s Cellular Automata models dv α (t) dt = v0 α + ξ α (t) v α (t) τ α + f α,α 1 (t), (1) where f α,α 1 describes the effect of α 1 on α, and is generally a function of relative velocity, v α (t), the velocity of α, and the headway, d α (t) = x α 1 (t) x α (t), or clearance, s α (t) = d α (t) length α Cellular Automata Cellular automata models are less detailed than the follow the leader models above. A basic approach is to divide the road into cells of equal length, x, divide the time into intervals of equal duration, t,allow each cell to be either occupied or vacant, x and set each car s speed to v i = ˆv i t, where ˆv i is an integer v max ˆ. At each timestep, the state of the cells, occupied or vacant, changes based on a set of rules crafted to simulate the movement of cars Macroscopic Models Unlike Microscopic models, Macroscopic models only deal with collections of vehicles. The calculations are done in terms of descriptions of these collectives: spatial vehicle density ρ(x, t), average velocity V (x.t), and traffic flow or flux Q(x, t) = ρ(x, t)v (x, t). The oldest and still most popular macroscopic model is by Lighthill and Whitham. This model is based on the observation that, away from ramps and other roads, the number of cars within a road is conserved. This leads to a continuity equation, ρ(x, t) t + Q(x.t) dx = 0. (2) 2

3 Figure 3: At the time of minimal distance between B and A, t min, the two agents do not intersect. Therefore, action u is not in the control obstacle. 3 Approach 3.1 Control Obstacles Our approach is based on the idea of a control obstacle. This is the set of controls that would lead to a collision with an obstacle at some point in the future. We can formulate this as follows. Given an obstacle B, let us denote the position it will have at time t by B(t). Given the position of agent A at time t = 0 and an action u, let us denote the position agent A will have after undertaking u for time interval t by A(t, u). We can now define the velocity obstacle generally as CO A B = {u t > 0 : A(t, u) B(t) }. (3) An example of a control u that is outside the control obstacle is given in Figure3. We can use collision detection to check the legality of proposed controls. Once a control sample is taken, we generate a space-time swept volume of the car. This volume is collided with all current trajectories of other agents and obstacles. Of the free controls, the one bringing the agents closest to the goal is chosen. The space-time swept volume is generated from the kinematic model of the car, shown in Figure 4. Let (x, y) be the position of the robot, θ its orientation, u S the speed, u φ the turning angle, and L the length be 1. Following [7], its kinematic constraints are given as Figure 4: The kinematic model of a car. x (t) = u s cos θ(t), (4) y (t) = u s sin θ(t), θ (t) = u s tan u φ By integrating the above equations (4), we can derive a formula for the position of a car at time t under the assumption that the actions of speed, u s, and steering angle, u φ, remain constant: ( 1 tan(u φ ) sin(u ) s tan(u φ )t) A(t, u) = 1 tan(u φ ) cos(u s tan(u φ )t) + 1 tan(u φ ) (5) We will assume that there are an arbitrary number of obstacles B i and that they are also subject to simple car kinematics. The space-time volume, shown in Figure 5, is created based on a constant control, time limit, and the agent kinematic model. These volumes are checked for collisions to determine if the agents controls will cause a collision within the time horizon, shown in Figure Goal Seeking We use an optimization procedure to navigate the robot among multiple moving obstacles, given in Algorithm 7. Let u be the action the robot would select if no moving obstacles were around, for instance the action that would lead the robot most directly towards its goal. We refer to u as the preferred action. The actual action u to give the robot is given. 3

4 Figure 5: Space-time volume for a car moving in a circle. Figure 7: The navigation algorithm. by the solution to the problem u = arg min u S u u. (6) B CO A Bi i 3.3 Intersection Behavior Road and Lane Boundaries The road lane boundaries are defined by static obstacles, is in Figure Traffic Lights We want the cars to stop at a light if the light is red and go if it s green. We also want them to continue on if they re in the intersection when the light turns red. Our approach is to model the light as a finite state machine with states, Figure 6: Space-time volumes for a few cars. {NS, EW, NS-Left-Turns, EW-Left-Turns, All-Stop i } (7) where each allows a certain flow of traffic, and the All-Stop i states signify that all traffic must stop (there is one state for each transition). 4

5 3.3.5 Slow to Start Rule Our approach also incorporates a slow to start rule meant to capture the lack of attention on the part of drivers. This rule states that a car with a speed of zero will only start moving during a timestep with a probability of p. The effect of this is random delay on the part of cars waiting in line Boundaries Figure 8: The static obstacles defining an intersection Automaton for Cars Each agent has a begining boundary and ending boundary. Based on the state of the light, the desired ending boundary of the agent will change to its final goal or to the end of the lane, depending on the state of the light. Additionally, each light state is associated with a few additional, temporary obstacles to help steer the cars. Once a car gets within the intersection, it can continue on to its final goal regardless of the state of the light Preference for Leading Car In general, cars should not react to the motion of cars that are behind them. To get the same behavior from our agents, we can simply check for each whether it is in front or behind. If the agent is behind, then it s trajectory volume is not checked against: n = o r (8) isbehind = (n L <= 0) (9) where o is the obstacle position, r is the robot position, and L is a segment along the robot s length. The boundaries of the simulation are clearly defined. The initial and final positions of robots can be easily mapped to a grid based traffic simulator to create a hybrid system. Currently, the robots are added with a probability q at their initial boundary when it is clear, and they are removed when they reach a grid cell around their final boundary. Rather than probabilistically adding robots, they could be added when a simulator signaled a car was incoming, and the information that the leading grid was occupied by a car with a certain velocity could be passed back. Likewise, the information that a car has reached the ending grid with a certain speed could be communicated to the grid based simulator were one connected. 4 Results As can be seen in Figure 9, 10, the simulator works well, simulating noisy traffic. The slow to start rule creates hestiant, unattentive drivers while the sampling method gives the cars the imperfect character of human drivers. Videos of this can be found at the website 1. 5 Conclusion This report has discussed an approach to simulating vehicle traffic at an intersection. The result is the desired noisy, human-like traffic that could add realism to media representations as well as offer insights int controller autonomous traffic systems. The approach is currently limited by its computational cost. Thus, future work will focus on optimizing the method by 1 wilkie/intersection.html 5

6 Figure 9: Cars crossing across the intersection and turning right. 1) precomputing the space-time volumes and 2) parallelizing the algorithm. Additional work could be done to bias the sampling and modify the space-time volumes to account for uncertainty. References [1] P. Fiorini and Z. Shiller. Motion planning in dynamic environments using the relative velocityparadigm. In 1993 IEEE International Conference on Robotics and Automation, Proceedings., pages , [2] P. Fiorini and Z. Shiller. Motion planning in dynamic environments using velocity obstacles. The International Journal of Robotics Research, 17(7):760, [3] E. Frazzoli, M.A. Dahleh, and E. Feron. Realtime motion planning for agile autonomous vehi- Figure 10: Cars turning left through the interseciton. cles. Journal of Guidance Control and Dynamics, 25(1): , [4] D. Hsu, R. Kindel, J.C. Latombe, and S. Rock. Randomized kinodynamic motion planning with moving obstacles. The International Journal of Robotics Research, 21(3):233, [5] K. Kant and S.W. Zucker. Toward efficient trajectory planning: The path-velocity decomposition. The International Journal of Robotics Research, 5(3):72, [6] J.P. Laumond, PE Jacobs, M. Taix, and RM Murray. A motion planner for nonholonomic mobile robots. IEEE Transactions on Robotics and Automation, 10(5): , [7] S.M. LaValle. Planning algorithms. Cambridge University Press, [8] J. Peng and S. Akella. Coordinating multiple robots with kinodynamic constraints along 6

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