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Administration. Feedback on assignment Late Policy Some Matlab Sample Code makeRobot.m, moveRobot.m. Introduction to Motion Planning. CSE350/450-011 16 Sep 03. Class Objectives. Cell decomposition procedures for motion planning Review exact approaches from last week

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• Feedback on assignment

• Late Policy

• Some Matlab Sample Code

• makeRobot.m, moveRobot.m

CSE350/450-011

16 Sep 03

• Cell decomposition procedures for motion planning

• Review exact approaches from last week

• Introduce approximating approaches

• A* Algorithm

• Introduction to potential field approaches to motion planning

• Motion Planning

• “Motion Planning Using Potential Fields”, R. Beard & T. McClain, BYU, 2003 *** PRINT THIS OUT FOR NEXT TIME ***

• “Robot Motion Planning”, J.C. Latombe, 1991

Given an initial robot position and orientation in a potentially cluttered workspace, how should the robot move in order to reach an objective pose?

• Visibility Graph Methods

• Cell Decomposition:

• Exact Decomposition

• Approximate: Uniform discretization & quadtree approaches

• Potential Fields

GOAL

START

GOAL

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START

The Visibility Graph Method (cont’d)

START

The Visibility Graph Method (cont’d)

We can find the shortest path

using Dijkstra’s Algorithm

GOAL

Path

START

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Exact Cell Decomposition Method

GOAL

START

1) Decompose Region Into Cells

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Exact Cell Decomposition Method (cont’d)

GOAL

START

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Exact Cell Decomposition Method (cont’d)

GOAL

START

• Construct Channel from shortest cell path

• (via Depth-First-Search)

GOAL

START

4) Construct Motion Path P from channel cell borders

• Perform cell tessellation of configuration space C

• Generate the cell graph G(V,E)

• Each cell in Cfree corresponds to a vertex in V

• Two vertices vi,vj V are connected by an edge eij if they are adjacent (8-connected for exact)

• Edges are weighted by Euclidean distance

• Find the shortest path from vinit to vgoal

Approximate Cell Decomposition

Uniform

r

Neighbors

Cell Decomposition Issues

• Explores G(V,E) iteratively by following paths originating at the initial robot position vinit

• For each OPEN node, only keep track of its minimum weight path from vinit

• The set of all such paths forms a spanning treeT  Gof the vertices OPEN so far

• Define a cost function f(v) = g(v) + h(v) where

• g(v) is the distance from vinit to v

• h(v) is a heuristic estimate of the distance from v to vgoal

• Define a cost function k(v1,v2) = g(v1) + distance(v1,v2)

So how do we find the shortest path?A* Algorithm [Hart et al, 1968]

• Define a list OPEN with the following operations

• insert(OPEN, v) inserts node v into the list

• delete(OPEN, v) removes node v from the list

• minF(OPEN) returns the node v of minimum weight f(v) from OPEN, and removes it from the list

• isMember(OPEN, v) returns TRUE if v is in the list, false otherwise

• isEmpty(OPEN) returns TRUE if the list is empty, false otherwise

• Define the following operations over our spanning tree T

• insert(T, v2, v1) inserts node v2 into the tree with ancestor v1

• delete(T, v) removes node v from the tree

A* Algorithm (cont’d)

A* - The Algorithm (Latombe, 1991)

function A*(G, vinit, vgoal, h, k)

insert(T, vinit, nil); % vinit is the tree root

insert(OPEN, vinit);

while isEmpty(OPEN)==FALSE

v = minF(OPEN); % optimal node from f(v) metric

if v==vgoal

break;

end

for vPlus  adjacent(v) % All of the cells adjacent to cell v

if vPlus is NOT visited

insert(T, vPlus, v); % Expansion of spanning tree

insert(OPEN, vPlus);

else if g(vPlus)>g(v)+k(v,vPlus) % Better way to reach vPlus if true

delete(T, vPlus);

insert(T, vPlus, v); % Change ancestor to reflect better path

if isMember(OPEN, vPlus)

delete(OPEN, vPlus);

end

insert(OPEN, vPlus); % Change f(v) value to reflect better path

end

end

end % End while loop

if v==vgoal

return the path constructed from vgoal to vinit in T

else

return failure; % We didn’t find a valid path

end

• The algorithm requires an admissible heuristich(v)

where h*(v) is the optimal path distance from v to vgoal

• For admissible h(v) A* is guaranteed to generate a minimum cost path from vinit to vgoal for the given cell decomposition

• QUESTION: What would be a reasonable function choice for h(v)?

• When h(v)=0 (a “non-informed” heuristic), A* reduces to our friend Dijkstra’s Algorithm

The Optimality of A*

No Obstacles

Single Obstacle

Multiple Obstacles

No Path

• PROS

• Applicable to general obstacle geometries

• With A*, provides shorter paths than exact decomposition

• CONS

• Performance a function of discretization resolution (δ)

• Inefficiencies

• Lost paths

• Undetected collisions

• Number of graph vertices |V| grows with δ2 and A* runs in O(|V|2) time -> O(δ4) running time

Approximate Cell Summary

• Introduced by Khatib [1986] initially as a real-time collision avoidance module, and later extended to motion planning

• Robot motion is influenced by an artificial potential field – a force field if you will – induced by the goal and obstacles in C

• The field is modeled by a potential functionU(x,y) over C

• Motion policy control law is akin to gradient descent on the potential function

• A shortcoming of the approach is the lack of performance guarantees: the robot can become trapped in local minima

• Koditschek’s extension introduced the concept of a “navigation function” - a local-minima free potential function

Some Background…

Example Application: collision avoidance module, and later extended to motion planningTarget Tracking with Obstacle Avoidance

The Idea… collision avoidance module, and later extended to motion planning

GOAL

• In 2D, the gradient of a function collision avoidance module, and later extended to motion planningf is defined as

• The gradient points in the direction where the derivative has the largest value (the greatest rate of increase in the value of f)

• The gradient descent optimization algorithm searches in the opposite direction of the gradient to find the minimum of a function

• Potential field methods employ a similar approach

• Potential fields can live in continuous space collision avoidance module, and later extended to motion planning

• No cell decomposition issues

• Local method

• Implicit trajectory generation

• Prior knowledge of obstacle positions not required

• The bad: Weaker performance guarantees

Some Characteristics of Potential Field Approaches

Next Time…