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Robot Motion Planning Introduction One of the ultimate goals in robotics is to design autonomous robots: robots that you can tell what to do without having to say how to do it. Introduction

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introduction
Introduction
  • One of the ultimate goals in robotics is to design autonomous robots: robots that you can tell what to do without having to say how to do it.
introduction3
Introduction
  • To be able to plan a motion, a robot must have some knowledge about environment in which it is moving.
  • This motion planning problem has to be solved whenever any kind of robot wants to move in physical space.
  • The type of motions a robot can execute depend on its mechanics.
work space configuration space
Work Space & Configuration Space
  • R : robot, simple polygon
  • R (x, y): robot translated over a vector (x, y)
  • Pi : obstacles
  • S : S={P1,…,Pt}
work space configuration space5
Work Space & Configuration Space
  • R (x, y, Ф)
  • Ф: rotated clockwise through an angleФ
work space configuration space6
Work Space & Configuration Space
  • C(R) : configuration space, the parameter space of a robot R
  • The work space is the space where the robot actually moves around.
  • The configuration space is the parameter space of the robot.
work space configuration space7
Work Space & Configuration Space
  • Cforb (R,S) : forbidden configuration space
  • Cfree (R,S) : free configuration space
a point robot
A Point Robot
  • For a point robot, the work space and the configuration space are identical.
  • To simplify the description we restrict the motion of the robot to a large bounding box B that contains the set of polygons.
a point robot10
A Point Robot
  • This free space is a possibly disconnected region, which may have holes.
  • Our goal is to compute a representation of the free space that allows us to find a path for any start and goal position.
  • Use trapezoidal map tocompute free space.
a point robot11
A Point Robot
  • trapezoidal map (chapter 6)
  • O (nlogn)
a point robot14
A Point Robot
  • T(Cfree) : the trapezoidal map of the free space
  • Use T(Cfree) to find path from a start position pstart to a goal position pgoal.
a point robot18
A Point Robot
  • Any path we report must be collision-free, since it consists of segments inside trapezoids and all trapezoids are in free space.
  • We always find a collision-free path if one exists.
a point robot19
A Point Robot
  • Time Complexity:

process S (trapezoidal map) : O(nlogn)

breadth-first search : O(n)

minkowski sums
Minkowski Sums
  • The same approach can be used if the robot is a polygon.
  • The configuration-space obstacles are no longer same as the obstacles in work space.
  • CP : configuration-space obstacles
minkowski sums21
Minkowski Sums
  • CP : configuration-space obstacles
minkowski sums22
Minkowski Sums
  • Minkowski Sums
express the cp as minkowski sums
Express the CP as minkowski sums
  • p = (px, py)
  • -p = (-px, -py)
  • -S := {-p : p S }
  • Let R be a planar, translating robot and let P be an obstacle. Then the C-obstacle of P is

P (-R(0,0)).

minkowski sums25
Minkowski Sums
  • If P and R don’t have parallel edges, then the number of edges of the Minkoswi sum is exactly n + m.
compute minkowski sums
Compute Minkowski Sums
  • For each pair v, w of vertices, with v P and w R, compute v + w. Next, compute the convex hull of all these sums.
slide29
It only looks at pairs of vertices that are extreme in the same direction.
  • In the algorithm we use the notation angle(pq) to denote the angle that the vector pq makes with the positive x-axis.
slide30

c

P

3

4

R

1

2

a

b

slide31
i=1

j=1

v4=v1

w5=w1

Add (a+1) as a vertex to P R

(a+1)

slide32
Angle(v1v2)

Angle(w1w2)

angle(v1v2) = angle(w1w2)

i=2

j=2

a

b

1

2

(a+1)

slide33
i=2

j=2

Add (b+2) as a vertex to P R

(a+1)

(b+2)

slide34

c

Angle(v2v3)

Angle(w2w3)

angle(v2v3) > angle(w2w3)

i=2

j=3

b

3

2

(a+1)

(b+2)

slide35
i=2

j=3

Add (b+3) as a vertex to P R

(b+3)

(a+1)

(b+2)

slide36

c

Angle(v2v3)

Angle(w3w4)

angle(v2v3) < angle(w3w4)

i=3

j=3

b

4

3

(b+3)

(a+1)

(b+2)

slide37
i=3

j=3

Add (c+3) as a vertex to P R

(c+3)

(b+3)

(a+1)

(b+2)

slide38

c

Angle(v3v4)

Angle(w3w4)

angle(v3v4) > angle(w3w4)

i=3

j=4

(c+3)

a

4

3

(b+3)

(a+1)

(b+2)

slide39
i=3

j=4

Add (c+4) as a vertex to P R

(c+3)

(c+4)

(b+3)

(a+1)

(b+2)

slide40

c

Angle(v3v4)

Angle(w4w5)

angle(v3v4) < angle(w4w5)

i=4

j=4

(c+3)

(c+3)

(c+4)

a

4

(b+3)

(b+3)

1

(a+1)

(a+1)

(b+2)

(b+2)

slide41
i=4

j=4

Add (a+4) as a vertex to P R

(c+4)

(c+3)

(b+3)

(a+4)

(a+1)

(b+2)

slide42

(c+4)

(c+3)

(a+4)

(b+3)

(a+1)

(b+2)

time complexity
Time Complexity
  • It is O(n+m) if both polygons are convex.
  • It is O(nm) if one of the polygons is convex and one is non-convex.
  • It is O(n m ) if both polygons are non-convex.

2

2

translational motion planning46
Translational Motion Planning
  • A polygon with m vertices can be triangulated in O(mlogm) time.
  • Computing CP of each of triangles takes linear time in total.
  • Merge step can be done in O(nlogn).

T(n)=