1. Robotics meet Computer Science
Robot Motion Planning
Robotics meet Computer Science

2. Motion Planning

3. motion planning = collection of problems
Information required
Knowledge of spatial arrangement of workspace. E.g., location of obstacles
Full knowledge full motion planning
Partial knowledge combine planning and execution
motion planning = collection of problems

4. Motion problem The Problem: t is a continuous sequence of Pos’
Given an initial position POinit
Given a goal position POgoal
Generate: continuous path t from POinit to POgoal
t is a continuous sequence of Pos’

5. Work Space and Configuration Space
Reference Point
2 Degrees of freedom
3 Degrees of freedom

6. Components of the problem
A: single rigid object - the robot - moving in Euclidean space W (the wkspace).
W = RN, N=2,3
Bi , I=1,…,q. Rigid objects in W. The obstacles A B2 B1

7. The Environment of Motion Planning and Obstacle Avoidance

8. Configuration Space Idea
represent robot as point in space
map obstacles into this space
transform problem from planning object motion to planning point motion A B2 B1

9. Obstacles in C (cont.) Def: Connected Component
q1, q2 belong to the same connected component of Cfree iff they are connected by a free path
Objective of Motion Planning: generate a free path between 2 configurations if one exists or report that no free path exists.

10. Classic Path Planning Approaches
Roadmap – Represent the connectivity of the free space by a network of 1-D curves
Cell decomposition – Decompose the free space into simple cells and represent the connectivity of the free space by the adjacency graph of these cells
Potential field – Define a potential function over the free space that has a global minimum at the goal and follow the steepest descent of the potential function

11. Planning Approaches
1- Roadmap Captures connectivity of Cfree in a network of 1-D curves called “the roadmaps.” Once a roadmap is constructed: use a standard path. Roadmap Construction Methods: 1) Visibility Graph, 2) Voronoi Diagram, 3) Freeway Net and 4) Silhouette.

12. Roadmaps Examples qgoal qinit qinit qgoal
Visibility graph in 2D CS. Nodes: initial and final config + vertices of C-obstacles.
Voronoi diagram with polygonal C-obstacles

13. Decompose the free space into simple regions called cells
Cell Decomposition
Decompose the free space into simple regions called cells
Construct a non-directed graph representing adjacencies: the conectivity graph
Search for a path forming a “channel”
Two variations:
Exact: union of cells is exactly the free space
Approximate: union included in the free space

14. Cell Decomposition: Example10 1 2 3 4 5 6 7 8 9 11 12
qinit
qgoal 3 2 10 4 5 6 9 1 11 12 7 8

15. Trapezoidal Decomposition

16. Adjacency Graph Nodes: cells
Edges: There is an edge between every pair of nodes whose corresponding cells are adjacent.

17. Quadtree Decomposition

18. Octree Decomposition

19. Approximation Cell Decomposition
Now let us look at how C- obstacle query can be used for cell decomposition for path non-existence.
The cell decomposition will subdivide the configuration space into cells. If a cell lies entirely in C-obstacle, the cell is labelled as full cell. If … Otherwise, …
Here the full cells can be identified using C-obstacle query.
Configuration Space
full
mixed
empty

20. A potential field is a scalar function over the free space.
Potential Fields
A potential field is a scalar function over the free space.
To navigate, the robot applies a force proportional to the negated gradient of the potential field.
A navigation function is an ideal potential field that
has global minimum at the goal
has no local minima
grows to infinity near obstacles
is smooth

21. Attractive & Repulsive Fields
Slide 21

22. Problem Formulation for Point Robot
Input
Robot represented as a point in the plane
Obstacles represented as polygons
Initial and goal positions
Output
A collision-free path between the initial and goal positions

23. Breadth-First Search

24. Breadth-First Search

25. Breadth-First Search

26. Breadth-First Search

27. Breadth-First Search

28. Breadth-First Search

29. Breadth-First Search

30. Breadth-First Search

31. “Hey I got struck… I’ll choose another path”
A* Algorithm
“I believe this is this way takes me shortest to the destination…. Lets give it a try”
“Hey I got struck… I’ll choose another path”
Add all possible moves in an open list.
Make the best move as per open list status
Add all executed moves in the closed list

32. Genetic Algorithms “Show me some random paths so that I may decide”
“OK this path is the best to go till a point and this path the best for the other part of the journey… Let me mix them both…”
Generate random complete and incomplete solutions: source to nowhere, nowhere to goal and source to goal
Try to mix paths to attain optimality
Generate random paths between needed points

33. Chromosome - representation
An encoded expression of potential solutions. (usually in the form of string)
Path representation
Encode the order of visiting points as a string
Example

34. Chromosome - population
Population – a set of chromosomes
Chromosomes are generated randomly. i.e. the population contains a set of random solutions
Each robot is equipped with one population

35. Operator - Crossover
Exchange information from two selected chromosomes
Crossover point: the first identical node in both parents
Example:
Parent1:
Parent2:
Child1:
Child2:

36. Operator - Mutation Mutation rate is low (for escaping local optimum)
Randomly selects one gene (node) to mutate
Its following nodes are picked randomly from sequentially connected nodes

37. Evaluation - Fitness Fitness = Tour Length + Waiting Time
(Static) (Dynamic)

38. ANN with Back Propagation Algorithm
“Whenever this type of situation arrives… Always make this move”
“Hey rules failed… I’m struck… OK make random moves till you are out”
Frame input/output pairs for every situation comprising of robot position, goal position and environment
Learn these and use them in decision making
Make random moves when position deteriorates

39. Path Non-existence Problem
Initial
Robot
Obstacle
Obstacle
Goal
Slide 39

40. Path Non-existence Problem
Here is a slightly more complex example. We have 2 gear shaped obstacles in 2D plane. We need to move the gear-shaped robot from left to right through these two obstacles. The figure shows that the robot can be placed between these two obstacles. But can you quickly identify path non- existence for this problem?
% But it’s not intuitive to tell whether there is a collision-free path or no path exists for this example.
% In some sense, to claim the path non-existence is more difficulty than to find a collision free path.
% And demands more computations

41. Find planning path from o to x

42. initial position (1, 1), goal symbol
initial position (1, 1), goal symbol * and obstacles are marked by boxes