ASLAutonomous Systems Lab
|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
ASLAutonomous Systems Lab
Introduction | the see – think – act cycle
“position“
global map
Cognition
Path Planning
knowledge,
data base
mission
commands
Localization
Map Building
environment model
local mappath
|Autonomous Mobile RobotsMargarita Chli, Paul Furgale, Marco Hutter, Martin Rufli, Davide Scaramuzza, Roland Siegwart Introduction to Optimization Techniques 2
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
ASLAutonomous Systems Lab
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.
|Autonomous Mobile RobotsMargarita Chli, Paul Furgale, Marco Hutter, Martin Rufli, Davide Scaramuzza, Roland Siegwart Introduction to Optimization Techniques 3
[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.
ASLAutonomous Systems Lab
Introduction | the motion planning problem
|Autonomous Mobile RobotsMargarita Chli, Paul Furgale, Marco Hutter, Martin Rufli, Davide Scaramuzza, Roland Siegwart Introduction to Optimization Techniques 4
Goal
ASLAutonomous Systems Lab
Introduction | origins and historical developments
� Geometric optimization: Dido‘s problem
|Autonomous Mobile RobotsMargarita Chli, Paul Furgale, Marco Hutter, Martin Rufli, Davide Scaramuzza, Roland Siegwart Introduction to Optimization Techniques 5
ASLAutonomous Systems Lab
Introduction | origins and historical developments
� Functional optimization:
Brachistochrone Problem (1696) 1
� Posed as a riddle by Johan Bernoulli to prove his superiority over his brother
� Initially solved geometrically
|Autonomous Mobile RobotsMargarita Chli, Paul Furgale, Marco Hutter, Martin Rufli, Davide Scaramuzza, Roland Siegwart Introduction to Optimization Techniques 6
� Functional Optimization Formulation
[1]: J. Bernoulli. Problema Novum ad Cujus SolutionemMathematici Invitantur. Acta Eroditorum, page 269, 1696.
ASLAutonomous Systems Lab
Introduction | origins and historical developments
� Pontryagin‘s minimum principle 1
� Extension of variational calculus to problems where
|Autonomous Mobile RobotsMargarita Chli, Paul Furgale, Marco Hutter, Martin Rufli, Davide Scaramuzza, Roland Siegwart Introduction to Optimization Techniques 7
� Closed-form solutions restricted to
� Linear systems with quadratic cost function
� Simple non-linear problems
� Cannot easily treat obstacles
ASLAutonomous Systems Lab
Introduction | origins and historical developments
� 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
|Autonomous Mobile RobotsMargarita Chli, Paul Furgale, Marco Hutter, Martin Rufli, Davide Scaramuzza, Roland Siegwart Introduction to Optimization Techniques 8
Courtesy O. Khatib
ASLAutonomous Systems Lab
Introduction | origins and historical developments
� Dynamic Programming (DP) 1
� Bellman‘s principle of optimality
� Discretization entails curse of dimensionality
� Graph search
� Deterministic (forward searching) instances of DP
|Autonomous Mobile RobotsMargarita Chli, Paul Furgale, Marco Hutter, Martin Rufli, Davide Scaramuzza, Roland Siegwart Introduction to Optimization Techniques 9
� 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.
ASLAutonomous Systems Lab
1. Motion control
2. Local collision avoidance
3. Global search-based planning
Introduction | hierarchical decomposition
|Autonomous Mobile RobotsMargarita Chli, Paul Furgale, Marco Hutter, Martin Rufli, Davide Scaramuzza, Roland Siegwart Introduction to Optimization Techniques 10
ASLAutonomous Systems Lab
Introduction | work-space versus configuration-space
2θ
|Autonomous Mobile RobotsMargarita Chli, Paul Furgale, Marco Hutter, Martin Rufli, Davide Scaramuzza, Roland Siegwart Introduction to Optimization Techniques 11
Work-spaceConfiguration-spacex
y
Work-space
1θ
2θ
x
y
Configuration-space
1θ
ASLAutonomous Systems Lab
|Autonomous Mobile RobotsMargarita Chli, Paul Furgale, Marco Hutter, Martin Rufli, Davide Scaramuzza, Roland Siegwart Collision Avoidance 12
Motion Planning | Collision AvoidanceAutonomous Mobile Robots
Martin Rufli – IBM Research GmbH
Margarita Chli, Paul Furgale, Marco Hutter, Davide Scaramuzza, Roland Siegwart
ASLAutonomous Systems Lab
� 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
|Autonomous Mobile RobotsMargarita Chli, Paul Furgale, Marco Hutter, Martin Rufli, Davide Scaramuzza, Roland Siegwart Collision Avoidance 13
ASLAutonomous Systems Lab
� 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.
ASLAutonomous Systems Lab
Dynamic Window Approach (DWA) | working principle
v
|Autonomous Mobile RobotsMargarita Chli, Paul Furgale, Marco Hutter, Martin Rufli, Davide Scaramuzza, Roland Siegwart Collision Avoidance 15
ω
ASLAutonomous Systems Lab
� 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
|Autonomous Mobile RobotsMargarita Chli, Paul Furgale, Marco Hutter, Martin Rufli, Davide Scaramuzza, Roland Siegwart Collision Avoidance 16
yv
xv
ORRRO rrt +<+vp
Uτ
τ
≤≤
−=
t
RORORO
t
r
tDVO
0
,p
ASLAutonomous Systems Lab
� 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
|Autonomous Mobile RobotsMargarita Chli, Paul Furgale, Marco Hutter, Martin Rufli, Davide Scaramuzza, Roland Siegwart Collision Avoidance 17
yv
xv
ASLAutonomous Systems Lab
� The robot is assumed to move on piece-wise linear curves
� Identical to the Velocity Obstacles method, except that collision avoidance is
shared between interacting agents – fairness property
Reciprocal Velocity Obstacles | working principle
rp
|Autonomous Mobile RobotsMargarita Chli, Paul Furgale, Marco Hutter, Martin Rufli, Davide Scaramuzza, Roland Siegwart Collision Avoidance 18
yv
xv
Uτ
τ
≤≤
−=
t
RORORO
t
r
tDVO
0
,p
ASLAutonomous Systems Lab
Reciprocal Velocity Obstacles | working principle
� The robot is assumed to move on piece-wise linear curves
� Identical to the Velocity Obstacles method, except that collision avoidance is
shared between interacting agents – fairness property
|Autonomous Mobile RobotsMargarita Chli, Paul Furgale, Marco Hutter, Martin Rufli, Davide Scaramuzza, Roland Siegwart Collision Avoidance 19
yv
xv
ASLAutonomous Systems Lab
|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
Martin Rufli – IBM Research GmbH
Margarita Chli, Paul Furgale, Marco Hutter, Davide Scaramuzza, Roland Siegwart
ASLAutonomous Systems Lab
� 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
|Autonomous Mobile RobotsMargarita Chli, Paul Furgale, Marco Hutter, Martin Rufli, Davide Scaramuzza, Roland Siegwart Potential Field Methods 21
Neumann Dirichlet
ASLAutonomous Systems Lab
� 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
|Autonomous Mobile RobotsMargarita Chli, Paul Furgale, Marco Hutter, Martin Rufli, Davide Scaramuzza, Roland Siegwart Potential Field Methods 22
Neumann Dirichlet
Court
esy A
. M
asoud