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3.1 Path integrals and quantum mechanics
Since the mathematical construction of the path integral plays a central role in this work,
it would have been a severe gap if this work did not include an introduction to path
integrals and their use for the mathematical representation of quantum phenomena in
physics. Therefore, in the next two subsections, we discuss the concept of least action
in classical mechanics and its generalization to quantum mechanics via the use of the
path integral. Moreover, we provide the connection between the path integral and the
Schr˝odinger equation, one of the most important equations in quantum physics.
The Schr˝odinger equation was discovered in 1925 by the physicist and theoretical
biologist, Erwin Rudolf Josef Alexander Schr˝odinger (Nobel-Lectures 1965). The initial
idea of the path integral goes back Paul Adrien Morice Dirac, a theoretical physicist who
together with Schr˝odinger was awarded the Nobel Prize in Physics in 1933 for their work
on discovery of the new productive forms of atomic theory. Richard Phillips Feynman
(Nobel-Lectures 1972), also a theoretical physicist and Nobel price winner in 1965 for his
work on quantum electrodynamics, completed the theory of path integral in 1948.
3.1.1 The principle of least action in classical mechanics and the
quantum mechanical amplitude.
Let us assume the case where a dynamical system moves from an initial state xA to a
terminal state xB. The principle of least action (Feynman & Hibbs 2005) states that the
system will follow the trajectory x"A, x"1
, ..., x"N−1, x"B that is the extremum of the cost
function:
63
Object Description
| Title | Iterative path integral stochastic optimal control: theory and applications to motor control |
| Author | Theodorou, Evangelos A. |
| Author email | etheodor@usc.edu; theo0027@umn.edu |
| Degree | Doctor of Philosophy |
| Document type | Dissertation |
| Degree program | Computer Science |
| School | Viterbi School of Engineering |
| Date defended/completed | 2011-01-11 |
| Date submitted | 2011 |
| Restricted until | Unrestricted |
| Date published | 2011-04-29 |
| Advisor (committee chair) | Schaal, Stefan |
| Advisor (committee member) |
Valero-Cuevas, Francisco Sukhatme, Gaurav S. Todorov, Emo Schweighofer, Nicolas |
| Abstract | Motivated by the limitations of current optimal control and reinforcement learning methods in terms of their efficiency and scalability, this thesis proposes an iterative stochastic optimal control approach based on the generalized path integral formalism. More precisely, we suggest the use of the framework of stochastic optimal control with path integrals to derive a novel approach to RL with parameterized policies. While solidly grounded in value function estimation and optimal control based on the stochastic Hamilton Jacobi Bellman (HJB) equation, policy improvements can be transformed into an approximation problem of a path integral which has no open algorithmic parameters other than the exploration noise. The resulting algorithm can be conceived of as model-based, semi-model-based, or even model free, depending on how the learning problem is structured. The new algorithm, Policy Improvement with Path Integrals (PI2), demonstrates interesting similarities with previous RL research in the framework of probability matching and provides intuition why the slightly heuristically motivated probability matching approach can actually perform well. Applications to high dimensional robotic systems are presented for a variety of tasks that require optimal planning and gain scheduling.; In addition to the work on generalized path integral stochastic optimal control, in this thesis we extend model based iterative optimal control algorithms to the stochastic setting. More precisely we derive the Differential Dynamic Programming algorithm for stochastic systems with state and control multiplicative noise. Finally, in the last part of this thesis, model based iterative optimal control methods are applied to bio-mechanical models of the index finger with the goal to find the underlying tendon forces applied for the movements of, tapping and flexing. |
| Keyword | stochastic optimal control; reinforcement learning,; robotics |
| Language | English |
| Part of collection | University of Southern California dissertations and theses |
| Publisher (of the original version) | University of Southern California |
| Place of publication (of the original version) | Los Angeles, California |
| Publisher (of the digital version) | University of Southern California. Libraries |
| Provenance | Electronically uploaded by the author |
| Type | texts |
| Legacy record ID | usctheses-m3804 |
| Contributing entity | University of Southern California |
| Rights | Theodorou, Evangelos A. |
| Repository name | Libraries, University of Southern California |
| Repository address | Los Angeles, California |
| Repository email | cisadmin@lib.usc.edu |
| Filename | etd-Theodorou-4581 |
| Archival file | uscthesesreloadpub_Volume14/etd-Theodorou-4581.pdf |
Description
| Title | Page 77 |
| Contributing entity | University of Southern California |
| Repository email | cisadmin@lib.usc.edu |
| Full text | 3.1 Path integrals and quantum mechanics Since the mathematical construction of the path integral plays a central role in this work, it would have been a severe gap if this work did not include an introduction to path integrals and their use for the mathematical representation of quantum phenomena in physics. Therefore, in the next two subsections, we discuss the concept of least action in classical mechanics and its generalization to quantum mechanics via the use of the path integral. Moreover, we provide the connection between the path integral and the Schr˝odinger equation, one of the most important equations in quantum physics. The Schr˝odinger equation was discovered in 1925 by the physicist and theoretical biologist, Erwin Rudolf Josef Alexander Schr˝odinger (Nobel-Lectures 1965). The initial idea of the path integral goes back Paul Adrien Morice Dirac, a theoretical physicist who together with Schr˝odinger was awarded the Nobel Prize in Physics in 1933 for their work on discovery of the new productive forms of atomic theory. Richard Phillips Feynman (Nobel-Lectures 1972), also a theoretical physicist and Nobel price winner in 1965 for his work on quantum electrodynamics, completed the theory of path integral in 1948. 3.1.1 The principle of least action in classical mechanics and the quantum mechanical amplitude. Let us assume the case where a dynamical system moves from an initial state xA to a terminal state xB. The principle of least action (Feynman & Hibbs 2005) states that the system will follow the trajectory x"A, x"1 , ..., x"N−1, x"B that is the extremum of the cost function: 63 |
