16-745: Dynamic Optimization
Spring 2016
Instructor: Chris Atkeson, cga at cmu
TA: Akshara Rai, arai at andrew
MW 4:30-6 NSH 3002

The TA will hold office hourse from 3:30-4:30 Monday and Wednesday in NSH 4223.

Events of Interest

Course Description

• Jan 11: Introduction to the course.
  Goal: Introduce course.
  This years emphasis is TO BE DETERMINED
  Possibilities:
  • Robust control.
  • Robust learning.
  • What do you want? ...
• Jan 13: Function Optimization Example
  Goal: Introduce you to a useful tool, MATLAB and its optimization subroutines, and show you how to use them on an example.
  Robotics: redundant inverse kinematics. Using Matlab's fminsearch and fminunc. Using Matlab's fminsearch and fminunc, with desired posture. Using Matlab's fmincon. Relationship of Jacobian approach to gradient descent.
• Jan 18: No Class
• Jan 20: Handling 3D Orientation
  Goal: Enable you to do 3D robotics using optimization (and do the inverse kinematics assignment).
  Rotation matrices, Euler angles, and Quaternions. Metrics for how close two orientations are: Metrics for 3D Rotations: Comparison and Analysis, Rigid-Body Attitude Control: Using Rotation Matrices for Continuous, Singularity-Free Control Laws, Closed-Loop Manipulator Control Using Quaternion Feedback
  Rotation matrix for small rotations
• Jan 25: Function optimization using first and second order gradient methods
  Goal: Review gradient descent approaches.
  A nice chapter on function optimization techniques: Numerical Recipes in C, chapter 10 (2nd or 3rd edition, 2nd edition is electronically available for free under Obsolete Versions): Minimization or Maximization of Functions, This material from any other numerical methods book is also fine.
  Resources: Matlab fminunc, Numerical Recipes, GSL, AMPL, NEOS, software list 1, Useful software guide,
  gradient method, line search, conjugate gradient, conjugate gradient v2, quasi-Newton/variable metric methods, and Newton's method.
• Jan 25: A Biased History of Artificial Neural Networks
  Goal: Make gradient descent and the chain rule more interesting.
  History, More info, Perceptron, Sigmoid units, Rectifier units (ReLU), Vanishing Gradients
• Jan 27: Siyuan Feng: Optimization for Robot Control (and the DARPA Robotics Challenge)
  Goal: Case study in the use of optimization for robot control.
  Siyuan's web page with videos/papers, Chris's web page with videos/papers,
• Feb 1: Non-gradient ("derivative-free") function optimization methods:
  Goal: Review non-gradient approaches.
  hill climbing (including local search, local unimodal sampling, pattern search, random search, random optimization), Nelder Mead/Simplex/Amoeba method, Matlab fminsearch, simulated annealing, fit surfaces (for example Response Surface Methodology (RSM), Memory-based Stochastic Optimization, and Q2), evolutionary algorithms, genetic algorithms, and ...
  Paper: Derivative-free optimization: A review of algorithms and comparison of software implementations by Luis Miguel Rios and Nikolaos V. Sahinidis, Book: Introduction to Derivative-Free Optimization
• Feb 1: Covariance Matrix Adaptation Evolution Strategy.
  Goal: Understand currently popular state of the art method.
  See also Hansen web page. Example1, Ex2, Ex3, Ex4.
  
• Feb 3: Constraints.
  Goal: Understand how to best handle constraints.
  Soft/hard constraints, penalty functions, Barrier functions, Lagrange Multipliers, Augmented Lagrangian method, Interior point methods vs. Simplex methods vs. soft constraint methods,
• Feb 3: Quadratic Programming and Sequential quadratic programming,
  Goal: Understand QP components used in state of the art robot control.
  Matlab fmincon. SNOPT, CVXGEN
  
• Feb 3: Automatic differentiation
  Goal: Learn how taking derivatives is much easier than you thought.
  
• Feb 8: Dynamics and Numerical Integration
  Goal: Review "mental simulation".
  Continous time, discrete time. Euler integration, Forward and inverse dynamics. Linearization.
• Feb 8: Formulating trajectory optimization as function optimization.
  Goal: Use the tools we have so far to do trajectory optimization.
  Examples of formulating a trajectory optimization problem as a function optimization problem: Case Studies In Trajectory Optimization: Trains, Planes, And Other Pastimes, Robert J. Vanderbei
  Example use of AMPL
  A free trial version of AMPL is available from here.
  AMPL is also available for remote use through the Neos Server. Click on SNOPT/[AMPL Input] under Nonlinearly Constrained Optimization.
  Example use of Matlab: pend1-x-u, pend1-u, pend1-x
  Spacetime Optimization: Witkin paper text Witkin paper figures
  
• Feb 15: Use of splines in trajectory optimization.
  Goal: Force smooth solutions.
  Cubic Hermite spline. Quintic Hermite interpolation. Example 1,
  Collocation, Pseudospectral X.
• Feb: Policy optimization I: Use function optimization.
  Goal: Optimize feedback.
  What is a policy? Known in machine learning/reinforcement learning as policy search or refinement, ...
  slides See examples in CMA-ES section for policy optimization.
• Feb: Ways to robustify function optimization:
  Goal: Tricks of the trade.
  Problems: How choose method?, more of an art than a science, local minima, bad answers, discontinuities, redundant/rank deficient constraints, bad scaling, no formulas for derivatives, you are lazy, computational cost.
  Techniques: Levenberg Marquardt, Trust regions, line search, scaling and preconditioning, regularize parameters, soft constraints, sparse methods, Continuation Methods, Paper on continuation methods, Hand of God, allow constraint violations, add extra constraints,
  Matlab recommendations
• Feb: Dynamic Programming.
  Goal: This is what makes dynamic optimization special.
  Bellman equation,
  slides
• Feb: Linear Quadratic Regulator,
  Goal: An important special case.
  Riccati Equation, Differential Dynamic Programming
• Feb: Ways to reduce the curse of dimensionality
  slides
• Feb: Policy Optimization II: Optimization using model-based gradients
  slides
• Feb: Robustness Robustness to random disturbances, varying initial conditions, parametric model error, structural modeling error such as high frequency unmodelled dynamics, and model jumps (touchdown and liftoff during walking, for example). Monte Carlo trajectory/policy optimization.
• Feb: Receding Horizon Control (a.k.a. Model Predictive Control (MPC)).
• Feb: Robustness using Linear Matrix Inequalities Robustness to parametric uncertainty in the linear(ized) model. I can't find a good reference on robustness using linear matrix inequalities, but here is a tutorial on LMIs
• Feb: Robustness: Policy Optimization with Multiple Models. Monte-Carlo, DP, and DDP approaches to Multiple Models.
• Mar: State Estimation, Uncertainty Propagation: Gaussian Propagation (like Kalman Filter), Unscented (like Unscented Filter), Second Order Kalman Filter (See Kendrick below).
  Review of Gaussians slides
  State estimation slides
  Matlab Kalman filter example and minimum jerk trajectory subroutine.
  Example mobile robot Kalman filter slides
  
• March 7-11: No Class
• March: Robustness and state estimation: Linear-quadratic-Gaussian control (LQG), Separation principle, Certainty equivalence, Example of bad interactions, Loop Transfer Recovery (LTR), A paper on the topic, Policy optimization approaches.
• March: Dual Control. Simple example. Information state DP.
• March: Local Approaches to Dual Control/Stochastic DDP Information state trajectory optimization. Stochastic Control for Economic Models, David Kendrick, Second Edition 2002.
• March: A*-like algorithms: R*
• March: Avoiding obstacles using Sampling based methods: RRT, slides Projected RRT, RRT* slides video 1 video 2 LQR-RRT* Random Sampling DP
• March: Avoiding obstacles using gradient methods: CHOMP STOMP
• April 6: Handling contact: Posa Talk 1pm April 8 NSH ???,, Contact-Invariant Optimization Hands Legs
  Trajectory Optimization for Full-Body Movements with Complex Contacts
• April: Learning From Demonstration
• April: Reinforcement Learning: Model free policy optimization. Kober, J.; Peters, J. (2011). Policy Search for Motor Primitives in Robotics, Machine Learning, 84, 1-2, pp.171-203
• April: Comparison of various RL methods: CMA-ES, CEM, PI2. Freek Stulp and Olivier Sigaud. Path Integral Policy Improvement with Covariance Matrix Adaptation. In Proceedings of the 29th International Conference on Machine Learning (ICML), 2012.
• April: Trajectory optimization based on integrating the dynamics: calculus of variations, Euler-Lagrange equation, Discrete time Pontryagin's minimum principle, Pontryagin's minimum principle, Hamilton-Jacobi-Bellman equation, costate equations, shooting methods, multiple shooting methods, Karush-Kuhn-Tucker conditions Continuation Methods, Meta-optimization, Learning during optimization
• Apr. 25: Project presentations
• Apr. 27: Project presentations
• May ?: Project Writeups Due

Assignments

• Assignment 0 (Due Jan. 17): Send CGA email: Who are you? Why are you here? What research do you do? Describe any optimization you have done (point me to papers or web pages if they exist). Any project ideas? What topics would you especially like the course to cover?
  Be sure your name is obvious in the email, and you mention the course name or number. I teach more than one course, and a random email from robotlover@cs.cmu.edu is hard for me to process.
• Assignment 1 (Due Feb. 7): Using Optimization to do Inverse Kinematics
• Assignment 2 (Due April 30): Using LQR and DDP

Project suggestions