administration feedback on assignment late policy some matlab sample code –makerobot.m,...

Administration

• Feedback on assignment

• Late Policy

• Some Matlab Sample Code

– makeRobot.m, moveRobot.m

Introduction to Motion Planning

CSE350/450-01116 Sep 03

Class Objectives

• 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

Supporting References

• 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

The Basic Motion Planning Problem

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

Planning Approaches

• Roadmap Approach:

– Visibility Graph Methods

• Cell Decomposition:

– Exact Decomposition

– Approximate: Uniform discretization & quadtree approaches

• Potential Fields

The Visibility Graph Method

GOAL

START

The Visibility Graph Method (cont’d)

GOAL

START

The Visibility Graph Method (cont’d)

GOAL

START

The Visibility Graph Method (cont’d)

GOAL

START

The Visibility Graph Method (cont’d)

GOAL

START Path

We can find the shortest pathusing Dijkstra’s Algorithm

Exact Cell Decomposition Method

GOAL

START

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1) Decompose Region Into Cells

Exact Cell Decomposition Method (cont’d)

GOAL

START

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2) Construct Adjacency Graph

Exact Cell Decomposition Method (cont’d)

GOAL

START

3) Construct Channel from shortest cell path(via Depth-First-Search)

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Exact Cell Decomposition Method (cont’d)

GOAL

START

4) Construct Motion Path P from channel cell borders

Exact Cell Decomposition Method Using Euclidean Metric

GOAL

START

Exact Cell Decomposition Method Using Euclidean Metric

GOAL

START

1. Perform cell tessellation of configuration space C

– Uniform or quadtree

2. 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

3. Find the shortest path from vinit to vgoal

Approximate Cell Decomposition

Cell Decomposition

Uniform Quadtree

Cell Decomposition Issues

1. Continuity of trajectory a function of resolution

2. Computational complexity increases dramatically with resolution

3. Inexactness. Is this cell an obstacle or in Cfree?

r Neighbors

• 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 tree T G of 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 heuristic h(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*

)()(0 * vhvhVv

A* Simulations

No Obstacles Single Obstacle

A* Simulations (cont’d)

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

The Potential Field Approach

• 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 function U(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: Target Tracking with Obstacle

Avoidance

GOAL

The Idea…

• In 2D, the gradient of a function f 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

Flashback: What is the Gradient?

yy

fxx

ff ˆˆ

• Potential fields can live in continuous space

– 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

• Generating potential functions for motion control

• PRINT OUT COPY OF POTENTIAL FIELD NOTES by R. Beard (they’re on the web page).

Next Time…