1. EE631 Cooperating Autonomous Mobile Robots Lecture: 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 ri

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 a6:
A, B, Y are constant matrices calculated from boundary conditions
a6 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 qf from q0. 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 z1-z4 plane
indicates collision region within a circle of
radius ri+R+l/2, since

15. Dynamic Collision Avoidance Criteria
Time criterion + geometrical criterion + path parameterization
g2, g1i, g0i are analytic functions of their arguments and can be calculated real time
a6k exists if g2>0
g2>0 holds for every points except boundary points

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

17. Global Motion Planning
Algorithm flow chart

18. In 2D environment with static obstacles (a6=0)
Simulations
Static obstacles
Feasible trajectory
Start
Goal
In 2D environment with static obstacles (a6=0)

19. Collision Trajectory Circles are drawn with 5 second spacing
Moving obstacles
Robot
Static obstacles
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

20. Global Collision–Free Trajectory
Moving obstacles
Robot
Static obstacles
a61=9.4086*10-6, a62=4.9973*10-6

21. Global Collision–Free Trajectory
Moving obstacles
Robot
Static obstacles
Moving obstacle changes velocity:
Original velocity [-0.15,-0.1], new velocity [0.15,-0.29]
Calculated a62=9.4086*10-6, a62=4.9973*10-6

22. Readings:
“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 Page(s):