16-745: Dynamic Optimization
Spring 2017
Instructor: Chris Atkeson, cga at cmu
TA: Stefanos Nikolaidis snikolai at andrew
MW 4:30-6 NSH 3002

The TA will hold office hours Monday and Wednesday 3.30-4.30pm, Personal Robotics Lab, NSH 4502

Events of Interest

Course Description

• Jan 18: Introduction to the course.
  Goal: Introduce course.
  This years emphasis is What happens when multiple optimizers interact?
  
• Jan 18: 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 23: 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 30: 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
• 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
• Covariance Matrix Adaptation Evolution Strategy.
  Goal: Understand currently popular state of the art method.
  See also Hansen web page. Example1, Ex2, Ex3, Ex4.
  
• 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,
• Quadratic Programming and Sequential quadratic programming,
  Goal: Understand QP components used in state of the art robot control.
  Matlab fmincon. SNOPT, CVXGEN
  
• Automatic differentiation
  Goal: Learn how taking derivatives is much easier than you thought.
  
• Dynamics and Numerical Integration
  Goal: Review "mental simulation".
  Continous time, discrete time. Euler integration, Forward and inverse dynamics. Linearization.
• 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
  
• Use of splines in trajectory optimization.
  Goal: Force smooth solutions.
  Cubic Hermite spline. Quintic Hermite interpolation.
  Collocation, Pseudospectral X.
  Wavelets
• 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.
• 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
• Dynamic Programming.
  Goal: This is what makes dynamic optimization special.
  Bellman equation,
  slides
• Linear Quadratic Regulator,
  Goal: An important special case.
  Riccati Equation, Differential Dynamic Programming
• Ways to reduce the curse of dimensionality
  Goal: Tricks of the trade.
  slides
• Policy Optimization II: Optimization using model-based gradients
  Goal: The Chain Rule Is Powerful.
  slides
• Robustness
  Goal: How To Handle Bad Models.
  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.
• Robustness using Linear Matrix Inequalities
  Goal: Handling Parametric Uncertainty.
  Robustness to parametric uncertainty in the linear(ized) model. Tutorial on LMIs, Slides: Continuous time stability slide 47, Discrete time stability slide 51
• Receding Horizon Control (a.k.a. Model Predictive Control (MPC))
  Goal: Online Optimization.
  
• Robustness: Policy Optimization with Multiple Models.
  Goal: A powerful tool to handle all kinds of uncertainty.
  Monte-Carlo, DP, and DDP approaches to Multiple Models.
• Finding Better Ways To Do Task
  Goal: Think about an important current research problem.
  
• Bayesian Filters
  Goal: Explicitly model uncertainty.
  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
  
• Robustness and state estimation:
  Goal: How to combine state estimation and control.
  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.
• Dual Control. Simple example. Information state DP.
• Local Approaches to Dual Control/Stochastic DDP Information state trajectory optimization. Stochastic Control for Economic Models, David Kendrick, Second Edition 2002.
• A*-like algorithms: R*
• Avoiding obstacles using sampling-based methods: RRT, slides Projected RRT, RRT* slides video 1 video 2 LQR-RRT* Random Sampling DP
• Avoiding obstacles using gradient methods: CHOMP STOMP
• Learning From Demonstration
• 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
• 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.
• Inverse Reinforcement Learning. Abbeel slides Finn slides
• What's new?
  • Combine trajectory optimization (model-based) and policy learning (model-free). I did some work on this 20+ years ago. Now it is coming back.
    Robot Learning From Demonstration, ICML '97, (postscript),
    Learning tasks from a single demonstration, ICRA '97,
    Nonparametric Model-Based Reinforcement Learning, NIPS '97,
    Using Local Trajectory Optimizers To Speed Up Global Optimization in Dynamic Programming, NIPS 93
    Random Sampling of States in Dynamic Programming, Trans SMC, 2008
    Combining Model-Based and Model-Free Updates for Trajectory-Centric Reinforcement Learning
    
  • Create primitives and learn to combine them. (Libin Liu).
    Akihiko Yamaguchi
    Google semantic event chains
    Freek Stulp
    another Cheng paper
    
  • What did the Berkeley folks say?
    See second half of Sergey Levine's lecture
    Finn's lecture on Transfer, Learning to Learn
    See last slide of Abbeel's lecture
    
• Review of Traditional Approaches 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
• May 1: Project presentations
• May 3: Project presentations
• May ?: Project Writeups Due

Assignments

• Assignment 0 (Due Jan. 20): 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 Jan. 31): Using Optimization to do Inverse Kinematics
• Assignment 2 (Due Feb. 26): Using Function Optimization To Optimize A Policy (For Walking)
• Assignment 3 (Due Apr. 9): Discrete Optimization Using Continuous Optimization

Other relevant classes

• CMU's Deep Reinforcement Learning and Control
• Berkeley's Deep Reinforcement Learning

Project suggestions