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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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administration
Administration
  • Feedback on assignment
  • Late Policy
  • Some Matlab Sample Code
    • makeRobot.m, moveRobot.m
slide2

Introduction to Motion Planning

CSE350/450-011

16 Sep 03

class objectives
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
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
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
Planning Approaches
  • Roadmap Approach:
    • Visibility Graph Methods
  • Cell Decomposition:
    • Exact Decomposition
    • Approximate: Uniform discretization & quadtree approaches
  • Potential Fields
the visibility graph method cont d11
The Visibility Graph Method (cont’d)

We can find the shortest path

using Dijkstra’s Algorithm

GOAL

Path

START

exact cell decomposition method

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Exact Cell Decomposition Method

GOAL

START

1) Decompose Region Into Cells

exact cell decomposition method cont d

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Exact Cell Decomposition Method (cont’d)

GOAL

START

2) Construct Adjacency Graph

exact cell decomposition method cont d14

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Exact Cell Decomposition Method (cont’d)

GOAL

START

  • Construct Channel from shortest cell path
  • (via Depth-First-Search)
exact cell decomposition method cont d15
Exact Cell Decomposition Method (cont’d)

GOAL

START

4) Construct Motion Path P from channel cell borders

approximate cell decomposition

Perform cell tessellation of configuration space C

    • Uniform or quadtree
  • 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
cell decomposition
Cell Decomposition

Uniform

Quadtree

cell decomposition issues

Continuity of trajectory a function of resolution

  • Computational complexity increases dramatically with resolution
  • Inexactness. Is this cell an obstacle or in Cfree?

r

Neighbors

Cell Decomposition Issues
so how do we find the shortest path a algorithm hart et al 1968

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]
a algorithm cont d

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
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 optimality of a

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*
a simulations
A* Simulations

No Obstacles

Single Obstacle

a simulations cont d
A* Simulations (cont’d)

Multiple Obstacles

No Path

approximate cell summary

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
some background

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…
flashback what is the gradient

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?
some characteristics of potential field approaches

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
next time

Generating potential functions for motion control

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