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By continuing this process of splitting the paths from xA to xB into subpaths, the
path integral takes the form:
K(xA, xB) = lim
dt&0
"
...
" " NG
i=1
K
$
xi+1, xi
%
dx1dx2...dxN (3.4)
where the kernel K(xi+1, xi) is now defined as:
K(xi+1, xi) =
1
A
exp
.
i
¯h
*t L
$
xi+1 − xi
*t
,
xi+1 + xi
2
,
ti+1 + ti
2
%/
(3.5)
and A =
$
2#i¯hdt
m
%1/2
. The equations (3.4) and (3.5) above realize the path integral
formulation in discrete time. The path integral formulation is an alternative view of
Quantum Mechanics in the next section we discuss the Schr˝odinger equation and its
connection to path integral.
3.1.2 The Schr˝odinger equation
In this section, we show how one of the most central equations in quantum mechanics,
the Schr˝odinger equation, is derived from the mathematical concept of path integrals.
The connection between the two descriptions is of critical importance since it provides a
more complete view of quantum mechanics (Feynman & Hibbs 2005), but it is also an
example of mathematical connection between path integrals and PDEs.
The derivation starts with the waive function 4(x, t) which can be though as an
amplitude with the slight difference that the associate probability P(x, t) = |4(x, t)|2 is
the probability of being at state x at time t without looking into the past. Since the wave
function is an amplitude function it satisfies the integral equation:
68
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 82 |
| Contributing entity | University of Southern California |
| Repository email | cisadmin@lib.usc.edu |
| Full text | By continuing this process of splitting the paths from xA to xB into subpaths, the path integral takes the form: K(xA, xB) = lim dt&0 " ... " " NG i=1 K $ xi+1, xi % dx1dx2...dxN (3.4) where the kernel K(xi+1, xi) is now defined as: K(xi+1, xi) = 1 A exp . i ¯h *t L $ xi+1 − xi *t , xi+1 + xi 2 , ti+1 + ti 2 %/ (3.5) and A = $ 2#i¯hdt m %1/2 . The equations (3.4) and (3.5) above realize the path integral formulation in discrete time. The path integral formulation is an alternative view of Quantum Mechanics in the next section we discuss the Schr˝odinger equation and its connection to path integral. 3.1.2 The Schr˝odinger equation In this section, we show how one of the most central equations in quantum mechanics, the Schr˝odinger equation, is derived from the mathematical concept of path integrals. The connection between the two descriptions is of critical importance since it provides a more complete view of quantum mechanics (Feynman & Hibbs 2005), but it is also an example of mathematical connection between path integrals and PDEs. The derivation starts with the waive function 4(x, t) which can be though as an amplitude with the slight difference that the associate probability P(x, t) = |4(x, t)|2 is the probability of being at state x at time t without looking into the past. Since the wave function is an amplitude function it satisfies the integral equation: 68 |
