CSCI-6971: Mathematics for mobile robotics

CSCI-6971:
Mathematics for mobile robotics
Spring 2006

Wes Huang (whuang@cs.rpi.edu)
Kris Beevers (beevek@gmail.com)
Flip Lamoureux (lamoup@cs.rpi.edu)
RPI Algorithmic Robotics Laboratory

A number of useful mathematical tools have been applied to mobile robotics problems such as mapping, localization, navigation, perception, and so on. This seminar course will review three particularly applicable areas: linear algebra, probabilistic methods, and optimization techniques.

Logistics

Tuesdays and Thursdays, 4:00-5:30, MRC 334.
3 credits by permission of the instructor (Wes Huang)
One person is responsible for preparing lecture notes (to hand out) and a lecture for each session.
There will be no assignments or tests.
Everyone should attend all of the classes with very few exceptions.

Schedule

Class	Topics	Preparer
01/17	Linear algebra basics, Gaussian elimination, Gauss-Jordan elimination, inverses, LU decomposition, Crout's algorithm, rank, singular matrices	Wes
01/19	Vector properties, fundamental spaces of a matrix, eigen values and eigenvectors	Wes
01/24	Inverse existence/uniqueness, projections, solving inconsistent systems, bases, Gram-Schmidt orthogonalization, QR decomposition, pseudoinverses, SVD, properties of determinants	Wes
01/26	Eigenvalues and eigenvectors and their properties, matrix diagonalization, matrix exponentials, positive definite matrices	Wes
01/31	Properties of probabilities and random variables, convolutions, least squares estimation, Normal random variables, central limit theorem, Markov inequality, Chebyshev inequality, weak law of large numbers, Jensen's inequality, Chernoff bounds	Kris
02/02	General stochastic processes, Bernoulli process, Poisson process, random incidence paradox, Markov chains	Kris
02/07	Markov chains, Hidden Markov Models, Baum-Welch algorithm, Viterbi algorithm, training problem	Flip
02/09	Continuous HMM, null transition HMM, variable duration HMM, HMMs with sparse data, dynamic structure HMM, hybrid HMM	Flip
02/14	Markov decision processes (MDP), partial observability (POMDP), mobile robotics POMDP example	Flip
02/16	Monte Carlo integration, random number generation, variance reduction, antithetic variates, common random variates, control variates	Kris
02/21	Rao-Blackwellization, stratified sampling, importance sampling	Kris
03/07	Importance sampling, sequential importance sampling, sequential Monte Carlo	Kris
03/09	Double session: Sequential Monte Carlo, bootstrap/particle filter, sampling distributions, resampling strategies	Kris
04/06	Markov-chain Monte Carlo, Metropolis-Hastings, MCMC convergence, Gibbs sampling	Kris
04/26	Graphical models, belief propagation, loopy BP, nonparametric BP	Kris
03/14	Spring break
03/16	Spring break
03/23	Nonlinear optimization basics, univariate minimization	Flip
03/28	Steepest descent, convergence of steepest descent, conjugate gradient descent	Flip
04/04	Unconstrained optimization, random walk, pattern search, Powell's method, steepest descent, conjugate gradient, quasi-Newton methods, second-order methods, Wolfe's rule, Goldstein-Price	Flip
04/11	Constrained linear programming, Simplex, constrained nonlinear programming, exterior penalty function method, augmented Lagrange multiplier method	Flip
04/13	Simplex example, convergence rate, Levenberg-Marquardt, nonlinear least squares	Flip
04/18	Direct constrained optimization, sequential linear programming, sequential quadratic programming, generalized reduced gradient, sequential gradient restoration	Flip
04/13	Lagrange multipliers	Wes
04/20	Lagrange multipliers, quadratic programming with equality constraints, quadratic programming with inequality constraints, Karush-Kuhn-Tucker (KKT) condition	Wes