Fall 2005 1

Path Planning from text

Fall 2005 2

Outline

Point RobotTranslational RobotRotational Robot

Fall 2005 3

Visibility Graph (Point Robot)

start

goal

Edges between all

pairs of visible vertices

Use graph algorithm to find a path from start to goal

Fall 2005 4

Free Space (Point Robot)

Fall 2005 5

Path Planning (Point Robot)

Fall 2005 6

Path Planning (cont)

Fall 2005 7

Robot (translational)

polygonal

Fall 2005 8

C-space Obstacle of P

Fall 2005 9

Minkowski Sum Coordinate

dependent!

Fall 2005 10

Theorem

CP is P(-R(0,0))

R(0,0)

–R(0,0)

Proof: R(x,y) intersect P (x,y)P(-R(0,0))

Fall 2005 11

R(0,0)

–R(0,0)

PqyxRq

q

),(

onintersectiin point A

)0,0(, Ryqxq yx

)0,0(, Ryqxq yx

(x,y)

If intersect, (x,y) is in CPIf intersect, (x,y) is in CP

))0,0((,

,,

RPyx

yqxqqq

Pq

yxyx

q

Fall 2005 12

R(0,0)

–R(0,0)

))0,0((, RPyx

(x,y)

If (x,y) is in CP, R(x,y)&P intersectIf (x,y) is in CP, R(x,y)&P intersect

p

yy

xx

ryp

rxp

is,That

r

yyxx

yxyx

rprpyx

PppRrr

,,such that

, )0,0(,

Fall 2005 13

Computing Minkowski Sum

Fall 2005 14

Example

v1,v4

v2

v3

w1,w5 w2

w3

w4

[i,j] = (1,1)

Add v1+w1

angle(v1v2) > angle(w1w2)

j2

Fall 2005 15v1v2

v3

w1 w2

w3

w4

[i,j] = (1,2)

Add v1+w2

angle(v1v2) < angle(w2w3)

i2

Fall 2005 16v1v2

v3

w1 w2

w3

w4

[i,j] = (2,2)

Add v2+w2

angle(v2v3) > angle(w2w3)

j3

Fall 2005 17v1v2

v3

w1 w2

w3

w4

[i,j] = (2,3)

Add v2+w3

angle(v2v3) < angle(w3w4)

i3

Fall 2005 18v4,v1

v2

v3

w1 w2

w3

w4

[i,j] = (3,3)

Add v3+w3

angle(v3v4) > angle(w3w4)

j4

Fall 2005 19v4,v1

v2

v3

w5,w1 w2

w3

w4

[i,j] = (3,4)

Add v3+w4

angle(v3v4) < angle(w4w5)

i4

Fall 2005 20v4,v1

v2

v3

w5,w1 w2

w3

w4

[i,j] = (4,4)

Add v4+w4

Fall 2005 21

Non-convex polygons

Fall 2005 22

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(n2m2) if both polygons are non-convex.

Fall 2005 23

Example 2

v4,v1

v2

v3

w5,w1

w2

w3

w4

[i,j] = (1,1)

Add v1+w1

angle(v1v2) < angle(w1w2)

i2

Fall 2005 24v4,v1

v2

v3

w5,w1

w2

w3

w4

[i,j] = (2,1)

Add v2+w1

angle(v2v3) > angle(w1w2)

j2

Fall 2005 25v4,v1

v2

v3

w5,w1

w2

w3

w4

[i,j] = (2,2)

Add v2+w2

angle(v2v3) > angle(w2w3)

j3

Fall 2005 26v4,v1

v2

v3

w5,w1

w2

w3

w4

[i,j] = (2,3)

Add v2+w3

angle(v2v3) < angle(w3w4)

i3

Fall 2005 27v4,v1

v2

v3

w5,w1

w2

w3

w4

[i,j] = (3,3)

Add v3+w3

angle(v3v4) > angle(w3w4)

j4

Fall 2005 28v4,v1

v2

v3

w5,w1

w2

w3

w4

[i,j] = (3,4)

Add v3+w4

angle(v3v4) < angle(w4w5)

i4

Fall 2005 29v4,v1

v2

v3

w5,w1

w2

w3

w4

[i,j] = (4,4)

Add v4+w4

Fall 2005 30

Rotational Robot

R (x, y, Ф)Ф: rotated anti-clockwise through an angle Ф

Fall 2005 31

Rotatonal Robot Motion Plan

Piano mover applet

Fall 2005 32

C-space of Rotational Robot

Fall 2005 33

Path Planning (Rotational Robot)

Each slice: R(0,0,i): obtain a roadmapProject all roadmap to get “intersection” – a pure rotation from i to j

Use a slight larger robot to ensure pure rotation won’t collide with obstacles

Fall 2005 34

Homework

P

R

[use the grid line to compute the result as accurate as possible]

1. Compute CP w.r.t. R2. Compute CP w.r.t. R’3. R and R’ are exactly the same

robot, differ only in reference point. Are CPs in 1 and 2 the same?

4. Do 1 and 2 obtain the same answer regarding to the intersection query? That is, the configuration shown left is reported as intersection in 1 & 2.

R’