1. EE631 Cooperating Autonomous Mobile Robots Lecture 5: Collision Avoidance in Dynamic Environments Prof. Yi Guo ECE Dept.

2. Plan A Collision Avoidance Algorithm A Global Motion Planning Scheme

3. Nonholonomic Kinematic Model Coordinate transformation and input mapping ( ,  are within (-  /2,  /2)): Chained form (after transformation):

4. Assumptions: The Robot 2-dimensional circle with radius R Knowing its start and goal positions Onboard sensors detecting dynamic obstacles

5. Assumptions: The Environment 2D environment with static and dynamic obstacles Pre-defined map with static obstacle locations known Dynamic obstacles represented by circles with radius r i

6. Problem Formulation: Trajectory Planning Find feasible trajectories for the robot, enrouting from its start position to its goal, without collisions with static and dynamic obstacles.

7. Feasible Trajectory in Free Space A family of feasible trajectories: Boundary conditions  In original coordinate:  In transformed coordinate:

8. Parameterized Feasible Trajectory Imposing boundary conditions, parameterization of the trajectory in terms of a 6 :  A, B, Y are constant matrices calculated from boundary conditions  a 6 increases the freedom of maneuver accounting for geometric constrains posed by dynamic obstacles

10. Steering Paradigm Polynomial steering: Assume T is the time that takes the robot to get to q f from q 0. Choose then

11. A quick summary System model: chained form Feasible trajectories: closed form parameterization Steering control: closed form, piecewise constant solution (polynomial steering) Next: Collision avoidance -- explicit condition based on geometry and time

12. Dynamic Collision Avoidance Criteria Time + space collision

13. Dynamic Collision Avoidance Criteria Time criterion:  Assume obstacle moves at constant velocity during sampling period  In original coordinate:  In transformed coordinate :

14. Dynamic Collision Avoidance Criteria Geometry criterion:  In original coordinate:  In transformed coordinate: Mapping from x-y plane to z 1 -z 4 plane indicates collision region within a circle of radius r i +R+l/2, since

15. Dynamic Collision Avoidance Criteria Time criterion + geometrical criterion + path parameterization  g 2, g 1i, g 0i are analytic functions of their arguments and can be calculated real time  a 6 k exists if g 2 >0  g 2 >0 holds for every points except boundary points

16. Global Path Planning Using D* Search A shortest path returned by D* in 2D environment Robot path Static obstacles Start Goal

17. Global Motion Planning Algorithm flow chart

18. Simulations In 2D environment with static obstacles ( In 2D environment with static obstacles (a 6 =0) Static obstacles Feasible trajectory Start Goal

19. Collision Trajectory – Circles are drawn with 5 second spacing – Onboard sensors detect:  obstacle 1: center [23,15], velocity [0.1,0.2]  obstacle 2: center [45,20], velocity [-0.1,-0.1] – Collisions occurs Robot Moving obstacles Static obstacles

20. Global Collision–Free Trajectory a 6 1 =9.4086*10 -6, a 6 2 =4.9973*10 -6 Robot Moving obstacles Static obstacles

21. Global Collision–Free Trajectory  Moving obstacle changes velocity: Original velocity [-0.15,-0.1], new velocity [0.15,-0.29]  Calculated a 6 2 =9.4086*10 -6, a 6 2 =4.9973*10 -6 Robot Moving obstacles Static obstacles

22. Readings: Laumond book Chapter 1 “A new analytical solution to mobile robot trajectory generation in the presence of moving obstacles”, by Zhihua Qu, Jing Wang, Plaisted, C.E., IEEE Transactions on Robotics, Volume 20, Issue 6, Dec. 2004 Page(s):978 - 993