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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
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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.
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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.
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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
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