EE631 Cooperating Autonomous Mobile Robots

Lecture: Collision Avoidance in Dynamic Environments

Prof. Yi GuoECE Dept.

Plan

A Collision Avoidance Algorithm A Global Motion Planning Scheme

Nonholonomic Kinematic Model

Coordinate transformation and input mapping(, are within (-/2,/2)):

Chained form (after transformation):

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

Assumptions: The Environment 2D environment with static

and dynamic obstacles Pre-defined map with static

obstacle locations known Dynamic obstacles

represented by circles withradius ri

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.

Feasible Trajectory in Free Space A family of feasible trajectories:

Boundary conditions In original coordinate:

In transformed coordinate:

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

Steering Paradigm

Polynomial steering:

Assume T is the time that takes the robot to get to qf from q0. Choose

then

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

Dynamic Collision Avoidance Criteria

Time + space collision

Dynamic Collision Avoidance Criteria

Time criterion: Assume obstacle moves at constant velocity during sampling

period In original coordinate:

In transformed coordinate :

))1(,[ 00 ss TktkTtt

Dynamic Collision Avoidance Criteria

Geometry criterion: In original coordinate:

In transformed coordinate:

Mapping from x-y plane to z1-z4 planeindicates collision region within a circle of radius ri+R+l/2, since

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

Global Path Planning Using D* Search

A shortest path returned by D* in 2D environment

Robot path

Static obstacles

Start

Goal

Cost function: ( is distance, is penalty on obstacles)ppf d d=r+ r

Global Motion Planning

Algorithm flow chartAlgorithm flow chart

Simulations

In 2D environment with static obstacles (In 2D environment with static obstacles (a6=0)

Static obstacles

Feasible trajectory

StartStart

GoalGoal

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

RobotMoving obstacles

Static obstacles

Global Collision–Free Trajectory

a61=9.4086*10-6, a6

2=4.9973*10-6

RobotMoving obstacles

Static obstacles

Global Collision–Free Trajectory

Moving obstacle changes velocity: Original velocity [-0.15,-0.1], new velocity [0.15,-0.29]

Calculated a62=9.4086*10-6, a6

2=4.9973*10-6

RobotMoving obstacles

Static obstacles

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. 2004 Page(s):978 - 993