Navigation Functions

1
Navigation Functions

• Presented by Ji Yeong Lee
• Exact Robot Navigation using Artificial Potential
  Functions by Rimon and Koditschek

2
Overview

• Motivation
• Potential Functions may have local minimum.
• Force Control with Bounded input
• Navigation Function in Sphere World
• Generalization to Star World
• Generalization to Forrest-of-stars from Star
  World

3
Potential Function Method

• Potential Function V FS -gt R
• Mininum at qd (goal)
• Large near the obstacles
• Potential Field ?V determines the control law
• Thus, can solve a path-planning, a trajectory
  planning and a control problem at once

4
Navigation Functions

• Problems
• Potential Function can have local minima
• ?V may be too large
• Define Navigation function such that
• It has a unique minimum at qd (goal)
• ?V is bounded

5
Navigation Functions

• ?FS -gt0,1 is navigation function if
• Smooth on FS (at least C2)
• Has unique minimum at qd (goal)
• Uniformly maximal on the boundary of FS (i.e. on
  the boundary of obstacles)
• A Morse Function Hessian of ? is nonsingular at
  the critical points
• Can be shown to exists for every smooth
  connected, compact manifold M with boundary, and
  qd ?M

6
Property of Nav. Function

• Known that
• M, F two free configuration space
• If ? M?0,1, a navigation function on M and
• hF?M, diffeomorphism,
• then ? ??h is a navigation function on F.

7
Sphere world

• Environment populated by disjoint spherical
  obstacles, bounded by a sphere
• Obstacle function ?i, Ciq ?i(q)?0
• ?o(q) -q-q02 ?02
• bounding sphere centered at q0, radius ?0
• ?i(q) q-qj2 - ?j2
• spherical obstacles, centered at qi, radius ?i
• Union of all obstacles
• Goal Position ?k(q) q-qd2k

qd
8
Conditioning Functions

• Goal define Nav. Fun from obstacle rep.
• Analytic Switch Bounding
• Analytic switch
• s(q,?)1 if ?(q)0 (I.e. at the boundary of
  obstacle)
• s(q,?)0 if ?(q)0 (I.e. at the goal)
• Sharpening

9
Nav. Fn. for Sphere World

• For sufficiently large k, ?k(q) is a navigation
  function

obstacles
Local min
goal
k4
k6
k3
k7
k8
k10
10
Nav. Fn. ?k(q), varying k
k3
k4
k6
k7
k8
k10
11
Star World

• Obstacles Disjoint set of star shaped sets
• Define diffeomorphism between star-world and
  sphere world
• Sphere world obstacle Ci
• Centered at the center of the star shaped set
• Contained entirely in the star shaped set

12
Star Shaped Set to Sphere

• Translated scaling map
• Maps a star-shaped set to a sphere
• Ti(q)?i(q) q - qipi,
• where,
• qi center of Ci,
• pi center of sphere, ? radius of sphere
• use piqi

13
Star World to Sphere World

• Analytical Switch for each obstacle Ci
• 1 on the boundary of Ci,
• 0 on the boundary of all other Cjs and at qd
• Lin. Combination of Translated Scaling
• Emulate Ti in a neighborhood of Ci
• Identity at qd
• Resemble Identity far away from Cis

14
Nav. Fn. In Star World

• h? is a diffeomorphism between sphere world and
  star world (for suff. large ?)
• Thus, given a navigation function ? in a
  sphere-world, ??h? also a navigation function.

15
More Complicated Spaces

• Decompose obstacles into unions of overlapping
  stars
• Define adjacent graph
• nodes center of the star-shaped set
• edges connects centers of overlapping stars

16
More Complicated Spaces

• Forest-of-trees
• Tree-of-stars
• Union of stars whose adjacency graph is tree
• No cycle is allowed
• Overlapping regions between two obstacles must a
  connected set
• Forest-of-trees
• Environment populated by trees-of-stars

17
Purging

• Purging leaves from forest-of-trees
• L all leaves from the all the tree-of-stars
• f? is a diffeomorphism
• Each operation of f? maps the boundary of the
  leaves to the boundary of the parents

(add Ti is different from Ti used in star to
sphere)
18
Nav. Fn. in Forest-of-stars

• f?n??f?2?f?1 forest-of-stars -gt star-world (n
  max depth of trees)
• g?h??f?n??f?2?f?1 diffeomorphism between
  forest-of-stars to sphere world
• ?k? g? Nav function in forest-of-stars

19
Summary

• Navigation Function in Sphere World
• Diffeomorphism between Sphere World and Star
  World
• Diffeomorphism between Forrest-of-stars and Star
  World

20
Old Slides
21
Nav. Fn. ?k(q), varying k
22
Potential Field

• Potential Field V determines the control law
• Solves a path-planning, a trajectory planning and
  a control problem at once

23
Navigation Functions

• Potential Function can have local minima
• Define Navigation function such that it has a
• ?FS -gt0,1 is navigation function if
• Smooth on FS (at least C2)
• Has unique minimum at qd (goal)
• Uniformly maximal on the boundary of FS (i.e. on
  the boundary of obstacles)
• A Morse Function Hessian of ? is nonsingular at
  the critical points
• Can be shown to exists for every smooth
  connected, compact manifold M with boundary, and
  x0 ?M

24
Navigation Functions

• Potential Function can have local minima
• Define Navigation function such that it has a
• ?FS -gt0,1 is navigation function if
• Smooth on FS (at least C2)
• Has unique minimum at qd (goal)
• Uniformly maximal on the boundary of FS (i.e. on
  the boundary of obstacles)
• A Morse Function Hessian of ? is nonsingular at
  the critical points
• Can be shown to exists for every smooth
  connected, compact manifold M with boundary, and
  x0 ?M

25
Transformation

• ??(x)x / (?x), ?gt0
• ?????
• ?????
• ?????

26
Sphere world

• Obstacle is represented as obstacle function ?i,
  s.t. obstaclei q ?i(q)?0
• Sphere world
• ??(q) q-qd2?, ? gt 0
• ?i(q) -q-q02 ?02 bounding sphere
• ?i(q) -q-qj2 ?j2 spherical obstacles
• -gt

27
Potential Function

• Planning in Configuration Space
• Potential Function FS -gt R
• Navigation Function Potential Function with a
  Unique Minimum
• Traditional approach vs. Nav. Function
• Traditional Approach Path Planning-gtTraj.
  Planning -gtRobot Control
• Nav. Function Approach t(p, dp/dt)-?V(p)d(p,
  dp/dt) The potential field dictates the control
  law