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|Autonomous Mobile RobotsMargarita Chli, Paul Furgale, Marco Hutter, Martin Rufli, Davide Scaramuzza, Roland Siegwart Introduction to Optimization Techniques 1

Motion Planning | Introduction to Optimization TechniquesAutonomous Mobile Robots

Martin Rufli – IBM Research GmbH

Margarita Chli, Paul Furgale, Marco Hutter, Davide Scaramuzza, Roland Siegwart


Introduction | the see – think – act cycle

“position“

global map

Cognition

Path Planning

knowledge,

data base

mission

commands

Localization

Map Building

environment model

local mappath


see-think-actraw data

Sensing Acting

Information

Extraction

Path

Execution

Mo

tio

n C

on

tro

l

Pe

rce

ptio

n

actuator

commands

Real World

Environment


Introduction | definitions

� Object

An object “is something material [i.e. an element in ] that may be perceived

by the senses” 1. The union of objects forms the complement to the empty, or

free-space.


[1]: Merriam-Webster. Object - Definition. Website, 2012http://www.merriam-webster.com/dictionary/object

� Agent

Decision-making objects are agents. They adhere to a system description.


Introduction | the motion planning problem


Goal


Introduction | origins and historical developments

� Geometric optimization: Dido‘s problem




� Functional optimization:

Brachistochrone Problem (1696) 1

� Posed as a riddle by Johan Bernoulli to prove his superiority over his brother

� Initially solved geometrically


� Functional Optimization Formulation

[1]: J. Bernoulli. Problema Novum ad Cujus SolutionemMathematici Invitantur. Acta Eroditorum, page 269, 1696.



� Pontryagin‘s minimum principle 1

� Extension of variational calculus to problems where


� Closed-form solutions restricted to

� Linear systems with quadratic cost function

� Simple non-linear problems

� Cannot easily treat obstacles



� Potential Fields

� Special time-invariant case of variational calculus where no system model is specified

� Re-invented in the 1980s based on simple attractive and repulsive force analogy


Courtesy O. Khatib



� Dynamic Programming (DP) 1

� Bellman‘s principle of optimality

� Discretization entails curse of dimensionality

� Graph search

� Deterministic (forward searching) instances of DP


� Examples include Dijkstra, A*, D*

� Design of underlying graph

� Difficult to construct system com-pliant graph structures

[1]: R. Bellman. Dynamic Programming. Princeton University Press, Princeton, NJ, 1957.


1. Motion control

2. Local collision avoidance

3. Global search-based planning

Introduction | hierarchical decomposition



Introduction | work-space versus configuration-space

2θ


Work-spaceConfiguration-spacex

y

Work-space

1θ

2θ

x

y

Configuration-space

1θ


|Autonomous Mobile RobotsMargarita Chli, Paul Furgale, Marco Hutter, Martin Rufli, Davide Scaramuzza, Roland Siegwart Collision Avoidance 12

Motion Planning | Collision AvoidanceAutonomous Mobile Robots




� Methods compute actuator commands based on local environment

� They are characterized by

� Being light on computational resources

� Being purely local and thus prone to local optima

� Incorporation of system models

Classic collision avoidance | overview



� Working Principle

� Environment represented as an evidence grid locally

� Reduction of the grid to a 1D histogram by tracing a dense set of rays emanating from the robot up to a maximal distance

� All histogram openings large enough for the robot to pass become candidates

� The direction with the lowest cost function G is selected

Vector Field Histogramm (VFH) | working principle

|Autonomous Mobile RobotsMargarita Chli, Paul Furgale, Marco Hutter, Martin Rufli, Davide Scaramuzza, Roland Siegwart

� The direction with the lowest cost function G is selected

� Properties

� Does not respect vehicle kinematics

� Prone to local minima

Court

esy B

ore

nste

inet

al.


Dynamic Window Approach (DWA) | working principle

v


ω


� The robot is assumed to move on piece-wise linear curves

� The Velocity Obstacle is composed of all robot velocities leading to a collision

with an obstacle before a horizon time

Velocity Obstacles (VO) | working principle

τ


yv

xv

ORRRO rrt +<+vp

Uτ

τ

≤≤

−=

t

RORORO

t

r

tDVO

0

,p



� The Velocity Obstacle is composed of all robot velocities leading to a collision

with an obstacle before a horizon time

Velocity Obstacles (VO) | working principle

τ


yv

xv



� Identical to the Velocity Obstacles method, except that collision avoidance is

shared between interacting agents – fairness property

Reciprocal Velocity Obstacles | working principle

rp


yv

xv

Uτ

τ

≤≤

−=

t

RORORO

t

r

tDVO

0

,p


Reciprocal Velocity Obstacles | working principle


� Identical to the Velocity Obstacles method, except that collision avoidance is

shared between interacting agents – fairness property


yv

xv


|Autonomous Mobile RobotsMargarita Chli, Paul Furgale, Marco Hutter, Martin Rufli, Davide Scaramuzza, Roland Siegwart Potential Field Methods 20

Motion Planning | Potential Field MethodsAutonomous Mobile Robots




� Robot follows solution to the Laplace Equation

� Boundary conditions, any mixture of

� Neumann: Equipotential lines lie orthogonal to obstacle boundaries

� Dirichlet: Obstacle boundaries attain constant potential

Harmonic Potential Fields | working principle

02

2

=∂

∂=∆ ∑

iq

UU


Neumann Dirichlet


� Robot follows solution to the Laplace Equation

� Boundary conditions, any mixture of

� Neumann: Equipotential lines lie orthogonal to obstacle boundaries

� Dirichlet: Obstacle boundaries attain constant potential

Harmonic Potential Fields | working principle

02

2

=∂

∂=∆ ∑

iq

UU


Neumann Dirichlet

Court

esy A

. M

asoud

motion planning | introduction to optimization techniques...graph search deterministic (forward...

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