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ROB310 Mathematics for Robotics, 2015 Prof. Angela Schoellig ROB310 Mathematics for Robotics Overview of the Topics Covered Objective The objective of this document is to give you an overview of the topics covered, the problem sets associated with each of the topics, and related reading material. Reference Material The reading list refers to the following books and online content: [1] Kreyszig, Erwin, “Advanced engineering mathematics,” Wiley, 2011. [online] [2] Siegwart, Roland, Nourbakhsh, Illah Reza and Scaramuzza, Davide, “Introduction to autonomous mobile robots,” MIT press, 2011. [online] [slides] [3] Aström, Karl J., and Wittenmark, Björn, “Computer-controlled systems: theory and design,” Courier Corporation, 2013. [online] [4] Solomon, Justin, “Numerical algorithms: methods for computer vision, machine learning, and graphics,” CRC Press, 2015. [online] [uploaded on Blackboard as Reference 1] [5] Ratliff, Nathan, “Linear algebra: lecture notes,” 2014. [uploaded on Blackboard as Reference 2] [6] Pishro-Nik, Hossein, “Introduction to probability, statistics, and random processes,” Kappa Research, LLC, 2014. [online] [7] Haug, A. J., “A tutorial on Bayesian estimation and tracking techniques applicable to nonlinear and non-Gaussian processes.” MITRE Corporation, McLean, 2005. [online] Course Topics An outline of the topics covered in this course is provided below. The left column shows the lecture number. # Topic Reading 1 2 Introduction and Motivation System Models Discrete-time and continuous-time systems, exact discretization for sampled systems. Numeric Methods Sources of numeric error, numeric stability, numeric methods for root finding (bisection, fixed-point iteration, Newton’s method, secant method). [2], p. 1-10 [3], Ch. 2.1-2.3 3 1 [4], Ch. 2.1-2.2, 8.1-8.2 Problem Set #1 #1 ROB310 Mathematics for Robotics, 2015 Prof. Angela Schoellig 4 5 6 7 8 9 10 11 12 Numerical Integration and Differentiation Integration methods (midpoint, trapezoidal and Simpson’s rule), differentiation methods (forward, backward and centered difference). Ordinary Differential Equations (ODEs) Time-stepping methods (Euler forward, Euler backward), numerical properties (explicit vs. implicit methods, accuracy: localized and global truncation error, numerical stability). Ordinary Differential Equations (ODEs) Time-stepping methods (trapezoidal, Runge-Kutta), multi-step methods (midpoint, Adams-Bashforth), consistence of ODE methods. Simulation Parameters, Introduction to Optimization Characteristic time constants of an ODE, rules of thumb for simulation parameters (including simulation time, step size), optimization problem definition. Unconstrained Optimization First-order and second-order optimality conditions, convexity. Numeric Solvers for Unconstrained Optimization Problems Newton’s method and golden section search for one-dimensional problems, gradient descent and Newton’s method for multivariate problems. Constrained Optimization Equality and inequality constraints, KKT conditions, sketch of relevant optimization algorithms. Convex Optimization Problems Characteristics of convex problems. Advanced Linear Algebra Matrix properties (symmetric, positive (semi-)definite, condition number), vector and matrix norms, eigenvalue problem, singular value decomposition. 13 Singular Value Decomposition Derivation, interpretation, applications. 14 15 ➣ Midterm Review Session Probability Review Discrete and continuous random variables, joint probability distributions, marginalization, conditioning, Conditional Probability, Bayes’ Theorem Independence and conditional independence, examples for Bayes’ theorem. Bayes’ Theorem, Expected Value and Variance, Change of Variables Interpretation of Bayes’ theorem in terms of estimating a quantity from multiple observations, definition of expected value and variance, functions of random variables. Bayesian Tracking Recursive estimator equations based on prior update and measurement update. Remarks on Computer Implementation of Probabilistic Approaches Sampling an arbitrary distribution. 16 17 18 19 2 [4], Ch. 14 [4], Ch. 15.3.2 #1 15.1- #2, 3 [4], Ch. 15.3.315.5 #2, 3 [4], Ch. 9.1 #2, 3, 4 [4], Ch. 9.2 #4 [4], Ch. 9.3-9.4 #4 [4], Ch. 10.1-10.3 #5 [4], Ch. 10.4 #5 [4] Ch. 1.1-1.3, 4.2.1, 6.1-6.2; [5] Ch. I-II; [1] Ch. 7, 8 [4] Ch. 7.1-7.2; [5] Ch. III; [1] Ch. 7, 8 #5 #5 #6 [1], Ch. 24.1-24.3, 24.5-24.6, 24.9 [6] Ch. 1-3 [6] Ch. 1-6 #7 [6] Ch. 1-6 #8 [7] (partially) #9 [6] Ch. 12 #9 ROB310 Mathematics for Robotics, 2015 Prof. Angela Schoellig 20 21 22 Extracting Estimates from Probability Distributions Maximum Likelihood, Maximum A Posteriori, Minimum Mean Squared Error. Gaussian Probability Density Functions Marginalization, conditioning, summation, multiplication, passing through a nonlinearity. Complex Analysis Representations of complex numbers, complex conjugate, addition, subtraction, multiplication, division, complex functions and derivatives. 3 [6] Ch. 9.1 [6] Ch. 5.3.2 [1] Ch. 13 #9 4.2.3, # 10 # 10 ROB310 Mathematics for Robotics, 2015 Prof. Angela Schoellig ROB310 Mathematics for Robotics Syllabus and Course Information, Fall 2015 Objective The course addresses advanced mathematical concepts that are particularly relevant for robotics. Topics include optimization, advanced probability theory, complex analysis, advanced linear algebra, and numerical methods. Concepts will be studied in a mathematically rigorous way but will be motivated by robotics examples. The objective of this course is to provide a solid foundation of the mathematical concepts and methods used in the subsequent robotics courses (for example, Control Systems, Mobile Robotics and Perception, Computer Vision, and Machine Learning). The mathematical tools covered in this course are fundamental for understanding, analyzing, and designing robotics algorithms that solve tasks such as robot path planning, robot vision, robot control, and robot learning. Instructor Prof. Angela Schoellig, [email protected], Institute for Aerospace Studies (UTIAS), phone 416-667-7518 Teaching Assistant Max Cirtwill, [email protected] Schedule • Lecture: Mon 16:00–18:00, BA1230; Thu 16:00–18:00, RS208. • Office hours: after class on Thursday 18:00–19:00 (please come to Prof. Schoellig right after class) and by appointment (please send an e-mail to Prof. Schoellig); additional office hours will be offered before the midterm and final exam. Website We will use Blackboard for course administration: http://portal.utoronto.ca/. Reference Material Since we cover a wide range of mathematical topics tailored to robotics, reference material will provided for each topic separately. A good reference book that provides a broad overview over advanced mathematical methods is: Advanced Engineering Mathematics (Tenth Edition) by Erwin Kreyszig, Wiley, 2011. 1 ROB310 Mathematics for Robotics, 2015 Prof. Angela Schoellig Hand-written class notes will be made available on Blackboard; notes will typically be available for download before the lecture. Matlab/Simulink will be used for some of the exercises. The software is available to students through ECF. Check! Course Topics and Syllabus A rough outline of the topics covered in class is provided below. Each week the fourth lecture hour will be dedicated to robotics examples that show how the learned content is applied in robotics. Week Dates 1 2 Sep 10 Sep 14,17 3 Sep 21,24 4 Sep 28, Oct 1 5 Oct 5,8 6 Oct 15 7 Oct 19,22 8 Oct 26,29 9 Nov 2,5 10 Nov 9,12 11 Nov 16,19 12 Nov 23,26 13 Nov 30, Dec 3 14 Dec 7 Topic Problem Set Introduction and Motivation Numeric Methods Integration, differentiation, solving nonlinear equations. Numerics for Ordinary Differential Equations Multi-step methods, stability and accuracy. Optimization Techniques Analytical optimization. Optimization Techniques Numeric methods. Linear Algebra Review Eigenvalues and eigenvectors, least squares. ➣ Thanksgiving Day on Oct 12 Advanced Linear Algebra Singular value decomposition, principal components analysis. ➣ Review Session on Oct 26 ➣ Midterm Exam on Oct 29 during regular class hours, covers Weeks 1–7 Probability Theory Review Bayes rule, maximum likelihood. Gaussian Distributions Multivariate Gaussians, mixture of Gaussians. Advanced Probability Concepts Graphical models, Markov decision process. Complex Numbers and Functions Representations of complex numbers, functions of complex variables. Complex Integration Cauchy’s integral theorem, argument principle. ➣ Review Session #1 due Sep 24 #2 due Oct 1 #3 due Oct 8 #4 due Oct 15 #5 due Oct 26 #6 due Nov 12 #7 due Nov19 #8 due Nov 26 #9 due Dec 3 #10 due Dec 7 Problem Sets Problem sets will be handed out at the beginning of each section and are due a week later. Late submissions will not be accepted and will result in zero points. Students must hand in their own solution but are encouraged to discuss the problems with their peers. Questions can also be asked in the office hours. 2 ROB310 Mathematics for Robotics, 2015 Prof. Angela Schoellig Grading Grades will be assigned according to the following scheme: • Problem Sets, 20% • Midterm Exam, 30% • Final Exam, 50% Allowed aid for the midterm exam is a single sheet of paper (letter size). Students may enter on both sides of the aid sheet any information they desire, without restriction, except that nothing may be affixed or appended to it. Allowed aid for the final exam are two (2) single sheets of paper (letter size) and a non-programmable calculator. “Education is not the filling of a pail, but the lighting of a fire.” – Plutarch 3