1. Mathematics & Path Planning for Autonomous Mobile Robots
Caroline Paulic
Math 480
April 1, 2013

2. What is path planning for autonomous mobile robots?
Mobile- The robot moves around its environment
Autonomous- No outside human intervention
Path planning- Determining a path from an initial configuration to a goal configuration such that the robot doesn't collide with any obstacles

3. Keep These Questions In Mind
How can the robot be represented?
How can the space be represented?
How can the robot reason with respect to its representation of space?

4. How can the robot be represented?
Point robot- Abstract the robot down to a single point
Can make up for the robot's radius by dilating all obstacles out by a distance equivalent to the robot's radius

5. So, how should we represent our space?

6. Representing Space: Discrete Case
The environment is broken up into discrete chunks
This can be done in a number of ways, but we won't be covering that today
Note that these chunks don't necessarily have to be of the same size or shape (depending on what representation technique is chosen)

7. Representing Space: Voronoi Diagram
Represent each obstacle as a point
Each line segment is equidistant between two obstacles
Nodes are equidistant between three obstacles

8. Representing Space: Continuous Case
Represented by a continuous plane in Rn for an n-dimensional configuration space
A configuration space represents all of the possible kinematic states of the robot
The configuration space has one dimension for every degree of freedom of the robot
Unlike the discrete case, this space is not divided up at all

9. How can we search our space in order to find a path from the start to the goal?

10. Searching A Discrete Space: Voronoi Diagram
From a start position, move
directly away from the
nearest obstacle until we
reach a point that lies on
one of the line segments
Follow the segment until we are in a spot such that we can move directly towards the goal while avoiding obstacles

11. Searching A Discrete Space: Graph Search
Take the environmental representation and make it into a graph
Each cell represents a node in the graph
If a robot can move from one cell to another, then those nodes are connected in the graph
Cells that would result in a collision are not included in the graph
Search the resulting graph
Now, our problem is reduced to finding an efficient path from the start node to the goal node

12. Graph Search: Depth-First
Searches deep within the first node before moving to the next

13. Graph Search: Breadth-First
Searches through every node on one level before moving down to the next level

14. Graph Search: Dijkstra's Algorithm
Similar to breadth-first search
Difference is that it keeps track of the cost associated with each edge traversal (only positive costs are considered)
Still exhaustively enumerates all possibilities, but then chooses the one with the least estimated cost

15. Graph Search: Best-First
Choose the open node that is closest to the goal
For obstacle free environments, this is efficient
In more complex environments, not so much...

16. Searching A Continuous State Space: Potential Fields
Robot is treated as a particle acting under the influence of a potential field
The robot is modeled as carrying an electric charge

17. Potential Field: Obstacles
Obstacles carry the same charge as the robot (repels)
Typically modeled as a barrier that rises to infinity as the robot approaches it
Example: Uobstacle=||q - obstacle||-2
q is the robot's state
obstacle is the point on the obstacle that is closest to the robot

18. Potential Field: Obstacles
We can then take the gradient of our function to obtain an artificial force field
*Recall that the gradient of a function gives us the direction in which the function is growing the fastest*

19. Potential Field: Goal
We want to model the goal field such that the robot is drawn towards the goal
Typically modeled to be parabolic
Example: Ugoal=||q - goal||2

20. Potential Field: Sum
We can sum the goal field and the obstacle fields to get the net artificial potential

21. Potential Fields: Local Minima
Problems can arise with local minima
Robot gets stuck
Example:
There are a few solutions to this problem:
Taking a random number of steps in one direction
Invoking a procedural planner, such as a wall follower
Increasing the potential when the robot visits a region, so it is repelled by previously visited regions
Others that we won't cover today

22. Sources
Computational Principles of Mobile Robotics (Second Edition), Dudek and Jenkin
Introduction to Algorithms (Third Edition), Cormen, Leiserson, Rivest, and Stein
server/depth-first-vs-breadth-first/
res/L11.html