Regarding Python level necessary for the course

Transcription

1 Logistics First two recitations (next two weeks) Python basics (installation, basic syntax, basic programming), optional Making models for 3D printing w/ Blender Will announce details through Sakai Regarding Python level necessary for the course First of all, Python is SIMPLE! We will only use basic Python features and basic/math libraries We will not do heavy GUI programming TAs (offices at CBIM): Han, Shuai office hour: Thursdays 2-3pm Tang, Wei office hour: Wednesdays 1-2pm Use for any (general) course related questions or any question to me regarding 460/560. Try not send to my other addresses they will be replied but at a slower pace

2 CS 460/560 Introduction to Computational Robotics Fall 2017, Rutgers University Lecture 02 Math. Foundations I Instructor: Jingjin Yu

3 Outline Sets, functions, and cardinality of Sets Open sets and continuous functions Group theory concepts Topological space concepts In particular, we will discuss how they are useful in the context of computational robotics

4 Set, Set Operations, and Venn Diagram A set is a collection of elements. Examples: {1, a, cup, π, } All natural numbers, N n-dimensional Euclidean spaces, R n Set operations Union: A B = {x x A x B} Intersection: A B = {x x A x B} Complement: A = {x x U x A} Difference: A B = {x x A x B} Symmetric difference: A B = A B A B Venn diagram U A B

5 More Sets and Functions Subset ( ): B A x B, x A Superset ( ): A B B A Powerset: P S = {A A S}, example: S = {1,2} P S = {, 1, 2, {1,2}} Functions To fully specify a function we write f: X Y, x f x x X, f x Y For f to be a function, x X, f x Y must be uniquely defined Ex: for f x = x 2, we write f: R R, x x 2 A function is surjective if f X = Y. Ex: f: R R + {0}, x x 2 A function is injective if x 1 x 2, f x 1 f(x 2 ) Ex: f: N N, x x 2 A function is bijective if it is both surjective and injective Ex: f: R R, x x 3 A B A B U

6 Cardinality of Sets Cardinality: essentially the size of a set = 0 1, 2 = 2 P 1,2 = 4 P S = 2 S N = ℵ 0 - the smallest infinite (cardinal) number, read Aleph 0 R = ℵ 1 - there are more real number than natural numbers Measuring the relative cardinality of sets A B if there exists an injective function f: A B If A B and B A, then A = B This means there is a bijective function between A and B Cardinality of rational numbers? Same as the set of natural numbers Cardinality of real numbers? Uncountably infinite (Cantor s diagonalization argument)

7 Why Sets? Sets (and related concepts) are the cornerstone of math Formalizes mathematical study, i.e., making math precise An example in the context of computer science: set cover U: the universe with n elements S = {s 1, s 2,, s m }, each s i U Question: find the smallest collection of s 1, s 2,, s m whose union is U Turns out this problem is very hard for computers NP-hard Connection to robotics: Minimum Constraint Removal A robot wants to go from q s to q g There are multiple obstacles blocking the way Q: remove the least number of obstacles to go to q g Turns out this is as hard as set cover So, yeah, robotics problems can be hard Kris Hauser, The minimum constraint removal problem with three robotics applications, WAFR, 2012

8 A (Related) Fun Result Can algorithms solve all problems? Algorithm (roughly): a (JAVA) program that takes in a (finite) input A program deciding whether a number n is prime Problem: a parametrized question that also has a (finite) input E.g., is n a prime number? Not all problems can be solved by algorithms The number of programs is countable (i.e., as many as the natural numbers) But there are more problems!

9 Open Sets, Closed Set, Boundary on R n In Euclidean spaces, by convention, a set X is open if for all x X, there exists ε > 0 such that B x, ε X. Ex: R is open, a, b R, a < b, a, b is open Ex: The set x, y x 2 + y 2 < 1 R 2 is open The union of any number of open set is open A set is closed if its complement is open A set may be neither open nor closed, e.g., (a, b] The closure of a set is the set plus all its limit points The closure of a set is closed, e.g. Cl(S) is always closed Also, Cl S = Cl Cl S The interior of a set S, denoted S o, is the union of all open sets in S The boundary of a set S, denoted S, is Cl S S o.

10 Continuous Functions A function f: R R is continuous around x 0 R if ε > 0, δ > 0, s.t. x B(x 0, δ), f x (f x 0 ε, f x 0 + ε). Readily generalize to f: R n R

11 Why Open Sets and Continuous Functions? Open set: for precisely describing and solving a problem Obstacles are generally modeled as closed sets (not always) The free space (the space with obstacles removed) is then open If our robot stays in the free space, then no collisions Robot trajectories are mostly continuous functions E.g., in 2D, f: 0, t R 2

12 Group Theory Concepts A set G together with an binary operation is a group if the following group axioms are satisfied Closed: a, b G, a b G Associative: a b c = a (b c) Identity: e G, a G, a e = e a = a Inverse: a G, b G s.t. a b = b a = e Examples? The set of integers under addition The set of positive rational numbers under multiplication

13 Why Groups? A mathematical field full of sad (but curious) stories! Niels Henrik Abel (Norwegian, ) Invented group theory! Proved no explicit algebraic solutions for quintic polynomials And many other fundamental contributions Very unlucky! Sent group theory paper to Gauss, Gauss tossed it into garbage Sent another seminar paper to Cauchy, Cauchy misplaced it Died at the age of 26! Then he got a letter appointing him Professor at University of Berlin Abel Prize is basically the Nobel Prize in math Évariste Galois (French, ) Also invented group theory (independently, no internet then) Galois theory (more general than Abel s work) Also many many other important work But this guy was very passionate Political activist, went to prison Then chose to dual with an army officer and died 20 years old! Niels Henrik Abel Évariste Galois Images from wikipedia

14 Why Groups? Seriously Many types of discrete and continuous spaces are also groups! Ex: The Rubik s cube It s a planning problem! Ex: R under addition Ex: The unit circle under rotation (cos α, sin α) α β (cos β, sin β) (cos(α + β), sin(α + β)) α + β Can also do this using matrix multiplication cos(α + β) sin(α + β) sin(α + β) cos(α + β) More on this when we do coordinate transformations = cos α sin α sin α cos α cos β sin β sin β cos β

15 Topological Space A set X and a collection Γ of subsets of X form a topological space if Γ and X Γ Arbitrary union of elements of Γ is again in Γ Finite intersection of elements of Γ is again in Γ Note: here, open sets are defined differently from earlier E.g., point set topologies (from Wikipedia) A set A is closed if X A is open Image from wikipedia

16 Topological Spaces on R The standard topology on R is the one with a, b R Γ for all a b. This is similar to what have done before Is 0, 1 open or closed? Closed, because, 0 (1, ) is open What about i=1 i, i + 1 i Alternatively, we can have Γ = {, R} This is the trivial topology on R Or, we can have Γ = n, n n R {, R}? Similar topologies can be defined for R n

17 Homeomorphism (I) Why study topology? One of the use is that it helps us to classify spaces Intuitively, which two of the following spaces are similar? The first and the third are both one dimensional What about those? All similar to the circle Homeomorphism: two spaces X and Y are homeomorphic if there is a continuous function F: X Y that is bijective

18 Homeomorphism (II) We can build a bijection f: X Y by sliding from one end to another on both lines What about X Y

19 Homeomorphism (III) One can build a series of homeomorphism to deform between two objects, that is, F t, t 0, 1, is a bijective continuous function with domain X F 0 X = X, F 1 (X) = Y This is also known as a deformation Classic example: coffee mug and donut Image from wikipedia

20 Topological Manifolds Roughly speaking, an n-dimensional topological manifold M is a space such that for x M, there exists a neighborhood U of x homeomorphic to R n 0-dimensional manifolds: discrete spaces 1-dimensional manifolds: S 1 a, b, R 2-dimensional manifolds: R 2, S 2, T 2,