1. Q UALITY M OTION P LANNING I N A D YNAMIC E NVIRONMENT August 2011

2. P ROJECT G OAL Generate a path for a Robot in a 2D Environment containing both Static & Dynamic obstacles. Few assumptions: The Robot has a maximum velocity and a Time limitation to reach destination. The dynamic obstacles movement is known and cyclic. The static Obstacles are simple polygons.

3. I MPLEMENTATION & DEVELOPMENT E NVIRONMENT The project was written in C++ using Visual Studio 2008, with support of the following software packages: MFC - we used this package to create our GUI. CGAL - for basic implementations & functions we used in our algorithms.

4. A LGORITHM O VERVIEW Based on PRM Zucker-Kant Algorithm Amending the Zucker-Kant Algorithms to achieve higher probability of solving problems

5. A LGORITHM – THE A SSEMBLY L INE The solution at each stage is represented by a directed graph The final path is a list of edges on graph The graph is passed to each of the stages of the algorithm Each stage modifies the graph and brings it closer to solution

6. A LGORITHM CHART MaBakerView User Output User Input RunAlgorithm Stage 1 Stage 2 Stage 5 A s s e m b l y L i n e Algorithm Support

7. B EGINNING

8. S TAGE 1 Sample points on completely free configurations (vertices on the graph) Use input to determine sample strategy Use CGAL to determine free configuration

10. S TAGE 2 Connect close points (edges on the graph) Make sure each edge is in almost free configuration Use CGAL to determine free configuration

12. S TAGE 3 Transform each edge found in stage 2 into a distance – time graph, considering moving obstacles Solve using the VGraph method If unsolvable – give edge an infinite weight If solvable – give weight indicating the time it takes to pass the edge If no moving obstacles on edge, just time when moving in max velocity

13. Edges with heavy weights Edges with light weights

14. T HE Z UCKER K ANT P LANE Let’s look at one segment of movement intersecting with one dynamic obstacle: R R O O O Block Start Block End

15. T HE Z UCKER K ANT P LANE In Time to distance on path plane, we get: Time Distance Block Start Block End R R O O O

16. T HE Z UCKER K ANT P LANE After setting a goal time, we can solve with vGraph: Time Distance Block Start Block End Dest Time R R O O O

17. S TAGE 4 Run Dijkstra on the graph and find the final path This is the path the robot is going to use, we still don’t know the velocity function However – it is guaranteed that this path is passable

19. S TAGE 5 Transform the path found in stage 4 into a distance – time graph, considering moving obstacles Calculate velocity function, which determine the progress velocities on the path we found in stage 4. This promises the shortest arrival time to destination.

20. E XAMPLES

21. W HAT DID WE LEARN ? New algorithms for path planning Working with complicated libraries – MFC, CGAL Analytical geometry when applied in a program Working on a big project

22. P OSSIBLE IMPROVEMENTS Improved sampling strategies Incremental algorithm Fix the case we fail (allow sampling almost free places) 3D?

23. A DDITIONAL R EADING K. Kant and S. Zucker. Toward efficient planning: the path-velocity decomposition. International Journal of Robotics Research, 5(3):72–89, 1986 Path Planning in Dynamic Environments, Jur Pieter van den Berg, May 1981