Structure and Synthesis of Robot Motion

Review of Probabilistic Methods

Subramanian RamamoorthySchool of Informatics

30 January, 2009

Why Probability matters in Robot Motion

1. You may be describing some variable whose value you don’t precisely know (e.g., location of a ball in a dark room)

2. You may be strategizing against an adversary whose intentions you do not know (e.g., that ball comes with a defender who will kick it around)

3. You may want to handle complex computations with limited resources (e.g., exact shortest path to goal or approximate?)

4. You may want to reason about classes of systems, adversaries or variable ranges– This gets you close to philosophical issues, sometimes required to

overcome complexity issues

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Objectives of this Lecture

• Eclectic sampling of topics (can’t possibly teach you all you need to know about probability in 45 mins!)

• Begin with basic notion of probability– Continuous and discrete versions– How to think about this slippery notion, when?

• Some words about uses– Sampling– Use in approximation and optimisation

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A Simple Example

Try an experiment:• Go out your front door in the morning and tabulate your

sightings of Male/Female persons• The probability that you will see a continuous sequence of

200 men is very small

• Unless, of course, the boy scouts are marching by on their annual parade – you don’t have 200 independent samples!

• A lot of motion problems can break seemingly sensible assumptions relatively easily – be careful!

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Small! Quite 10)( 6020021

Another Example: Bertrand’s Paradox

• You have a unit disk on the plane• Throw a chord at random• What is the probability that

(length of an equilateral inscribed triangle)Answer 1 (random endpoints method):

Answer 2 (random radius method):

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3l

l

3.0

30303

18060

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radius. a toperp. base with trianglelequilatera ofThink 5.0p

What is Going on?!

We have not been precise enough about problem formulation– What exactly is a “random chord”, anyway?– What is the probability distribution? Answer: It depends!

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Probability: The Axiomatic Definition

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etc. ,disjoint for )()(;1)(0;(A)yprobabilit as of think normally'' way wein the behaves that measure a is

onsintersectiunder closed be andset empty ,complement includemust setany is

]1,0[:sets ofset a - ) of (subsets events measurable of Collection :

experimentan of outcomes possible all set), (a space State :

iii AAA

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Think back to Bertrand’s paradox:They were all different problem formulations!

Need for a Softer Touch

• Enumerating contingencies is non-trivial in open-ended robotics problems

• So, in order to make quick decisions, we may use probability descriptions as a proxy for the real thing and try to update as we go along

• The hard part is mapping between beliefs and ODE dynamics or possibilities and paths in continuous spaces with obstacles!

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[Src: R. Jeffrey, Princeton]

Think Before You Use

• In any real robot motion problem, one has to carefully model the problem first

• Arbitrariness at this stage will not really get solved by black magic later on in terms of inference algorithms or learning methodologies

A last warning before we jump in - Clearly distinguish between:– Nondeterministic and Probabilistic – Uncertainty/nondeterminism about structure and parameters

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Use Case: Sampling (Randomized Algo.)

Sec 5.2 of LaValle’s book

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Use Case: Sequential Decision Making

Sec 10.1-3 of LaValle’s book

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