**Professor**: Michael Erdmann (me51).

**Teaching Assistants**:

**Emails:** AndrewID -at- andrew.cmu.edu (with "AndrewID" shown in parentheses above).

**Location**:

**Time**:

** Michael Erdmann's Office Hours**: Prefer after class (or by appointment, office is GHC 9203).

** TA Office hours**:

Copies of (handwritten)
lecture notes are available online here.

**1.**Solution of Linear Equations.**2.**Polynomial Interpolation and Approximation.**3.**Solution of Nonlinear Equations.**4.**Roots of Polynomials, Resultants.**5.**Approximation by Orthogonal Functions (includes Fourier series).**6.**Integration of Ordinary Differential Equations.**7.**Optimization.**8.**Calculus of Variations (with applications to Mechanics).**9.**Probability and Stochastic Processes (Markov chains).**10.**Computational Geometry.**11.**Differential Geometry.

This is a graduate course. You are thus expected to pursue ideas and topics discussed in this course on your own beyond the level of the lectures. My aim is to cover some of the easy early material quickly, then spend more detailed time on the later material. My goal throughout the course is to acquaint you with fundamental algorithms and mathematical reasoning, as well as give you some implementation experience.

The course grade will be determined by performance on assignments, participation in class, and a class project (see below for further details). Class assignments will entail solving some problems on paper or implementing some of the algorithms discussed in the course.

The term project should take about a month of work (40 hours) per person. It should pursue a mathematical topic in a robotics setting that is not otherwise covered in detail in the course. Ideally, the project should be connected to your research. If you are a first year graduate student, you should view the project as a springboard to research involvement.

- Understand the geometry of linear systems of equations, including null spaces, row spaces and column spaces. Understand eigenvalues, diagonalization, singular value decomposition, least-squares solutions. Apply these skills to infer rigid-body transformations from data.
- Learn how to model data using interpolation, linear regression, and higher-order approximation methods. Apply these skills to recognize objects from point-cloud data.
- Understand different methods for finding roots of equations, in one dimension and higher, including bisection, Newton's method, Müller's method. Develop skills to analyze convergence rates.
- Generalize finite-dimensional linear algebra concepts such as orthogonal bases to perform function approximation, including using Legendre polynomials and Fourier series to compute least-squares approximations. Learn about different function norms.
- Develop techniques for solving differential equations numerically.
- Understand optimization techniques such as gradient descent, conjugate gradient, Newton's method. Learn the basics of convex optimization. Implement obstacle-avoidance viewed as cost-minimization.
- Develop methods from the Calculus of Variations for higher-order optimization. Understand the Principle of Least Action. Apply these techniques to develop the Lagrangian dynamics of a robot manipulator.
- Learn basic computational geometry techniques, such as Convex Hull and Voronoi diagrams. Implement a geometric robot motion planner.
- Understand Markov chains from the perspective of eigenvalues and matrix representations.
- Understand basic constructs of Differential Geometry, in particular curvature and torsion of curves, along with the Frenet formulas.

In order to pass this course you must do all the work required. "Doing all the work" entails attending class, submitting correct solutions for assignments, and completing the required tasks for a project.

You must submit a reasonable attempt at a solution for
*every* problem on an assignment by that assignment's due date.
Assignment problems (or subproblems) are graded on a "minus, check,
plus" scale. Numerically, in Gradescope you will see a 0, 1, or 2,
respectively, for any given problem (or subproblem). Both a "1" and a
"2" count as 1 point toward your assignment completion, but a "2" is a
way for us to tell you that we think you did really nice work.

If you receive a "0" on any problem (or subproblem), you may and must resubmit a correct solution for that problem by the resubmission deadline for the assignment. (The resubmission deadline is not an alternate deadline for the original assignment; you need to attempt every problem by the original deadline.)

For the project, you should submit a project proposal, a project writeup, and present your project publicly. The project proposal is due near midsemester. It should be about one paragraph long and describe what you plan to do, why it is interesting, and cite prior work. Project writeups should be five pages long. Project writeups are due at the end of the last presentation. Project presentations will occur near the end of the term, depending on room availabilities.

Projects may be individual projects or team projects. Team projects must be commensurately larger in scope than single person projects. Projects that are used in more than one course need to be significantly more substantial than single course projects. A team project that is used in more than one course needs to be very significant. (That said, if the whole class would like to work on one massive research project together, that could be fantastic.)

(Please note: Not doing all the work as described above within the time frame allotted means you fail the course. Please be aware that you must receive at least a "1" on every problem/subproblem in order to pass the course. The resubmission process is a way for you to accomplish that task. Everyone should be able to pass the course. However, be careful to work steadily. Letting work pile up can put you in a situation where you may need to drop the course rather than fail. This rarely happens, but it does occur sometimes.)

**1.**W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling. Numerical Recipes in C. Cambridge University Press. (Any edition.)

**2.**G. Strang. Introduction to Applied Mathematics. Wellesley-Cambridge Press. 1986.

**3.**G. H. Golub and C. F. Van Loan. Matrix Computations. Johns Hopkins University Press. 1983.

**4.**S. D. Conte and C. de Boor. Elementary Numerical Analysis. Third edition. McGraw-Hill. 1980.

**5.**G. E. Forsythe, M. A. Malcolm, and C. B. Moler. Computer Methods for Mathematical Computations. Prentice-Hall. 1977.

**6.**D. G. Luenberger. Introduction to Linear and Nonlinear Programming. Addison-Wesley. 1973.

**7.**R. Weinstock. Calculus of Variations. Dover Publications. 1974. (Reprint of 1952 McGraw-Hill edition.)

**8.**R. Courant and D. Hilbert. Methods of Mathematical Physics. Volume I. John Wiley and Sons. 1989. (Reprint of 1953 Interscience edition.)

**9.**W. Yourgrau and S. Madelstam. Variational Principles in Dynamics and Quantum Theory. Dover Publications. 1979. (Reprint of a 1968 edition.)

**10.**F. P. Preparata and M. I. Shamos, Computational Geometry, Springer-Verlag, New York, 1985. (Corrected and expanded printing: 1988.)

**11.**J.-C. Latombe, Robot Motion Planning, Kluwer Academic Publishers, Boston, 1991.

**12.**W. Feller. An Introduction to Probability Theory and Its Applications. Volume 1. Third edition. John Wiley and Sons. 1968.

**13.**B. O'Neill, Elementary Differential Geometry, Academic Press, New York, 1966. 2nd Edition: 1997.

Take care of yourself. Do your best to maintain a healthy lifestyle this semester by eating well, exercising, avoiding drugs and alcohol, getting enough sleep and taking some time to relax. This will help you achieve your goals and cope with stress.

All of us benefit from support during times of struggle. You are not alone. There are many helpful resources available on campus and an important part of the college experience is learning how to ask for help. Asking for support sooner rather than later is often helpful.

If you or anyone you know experiences any academic stress, difficult life events, or feelings like anxiety or depression, we strongly encourage you to seek support. Counseling and Psychological Services (CaPS) is here to help: call 412-268-2922 and visit their website at https://www.cmu.edu/counseling/. Consider reaching out to a friend, faculty, or family member you trust for help getting connected to the support that can help.

If you or someone you know is feeling suicidal or in danger of self-harm, call someone immediately, day or night:

CaPS: 412-268-2922

Resolve Crisis Network: 888-796-8226

If the situation is life threatening, call the police:

On campus: CMU Police: 412-268-2323

Off campus: 911

Please do not record lectures or take images of the professor. University policy on this matter suggests the following formal statement:

No student may record any classroom activity without express written consent from the professor. If you have (or think you may have) a disability such that you need to record or tape classroom activities, you should contact the Office of Equal Opportunity Services, Disability Resources to request an appropriate accommodation.Thank you.