16745: Dynamic Optimization
Spring 2017
Instructor: Chris Atkeson, cga at cmu
TA: Stefanos Nikolaidis snikolai at andrew
MW 4:306 NSH 3002
The TA will hold office hours Monday and Wednesday
3.304.30pm, Personal Robotics Lab, NSH 4502
Events of Interest
Last year's course

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,
RigidBody Attitude Control: Using Rotation Matrices for Continuous, SingularityFree Control Laws,
ClosedLoop 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,
quasiNewton/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

Nongradient ("derivativefree") function optimization methods:
Goal: Review nongradient 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),
Memorybased Stochastic Optimization, and
Q2),
evolutionary algorithms,
genetic algorithms,
and ...
Paper:
Derivativefree optimization: A review of algorithms and comparison of software implementations by Luis Miguel Rios and Nikolaos V. Sahinidis,
Book: Introduction to DerivativeFree 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: pend1xu,
pend1u,
pend1x
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 CMAES 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 modelbased 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.
MonteCarlo, 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.
LinearquadraticGaussian 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 samplingbased methods: RRT,
slides
Projected RRT,
RRT*
slides
video 1
video 2
LQRRRT*
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, 12, pp.171203

Comparison of various RL methods: CMAES, 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.

Review of Traditional Approaches
Trajectory optimization based on integrating the dynamics:
calculus of variations,
EulerLagrange equation,
Discrete time Pontryagin's minimum principle,
Pontryagin's minimum principle,
HamiltonJacobiBellman equation,
costate equations,
shooting methods,
multiple shooting methods,
KarushKuhnTucker conditions
Continuation Methods,
Metaoptimization,
Learning during optimization

May 1: Project presentations

May 3: Project presentations

May ?: Project Writeups Due
Assignments