Page 107 |
Save page Remove page | Previous | 107 of 289 | Next |
|
small (250x250 max)
medium (500x500 max)
Large (1000x1000 max)
Extra Large
large ( > 500x500)
Full Resolution
All (PDF)
|
This page
All
|
By substituting back we will have:
!
*[xi − %(ti; xi−1, ti−1]
#
=
"
dw
(2,)n exp
$
jwT
$
x(ti) − x(ti−1) − f (x, t)d'
%%
× exp
$
−
1
2
wTB(x, t)B(x, t)Twdt
%
=
"
dw
(2,)n exp
$
jwTA
%
exp
$
−
1
2
wTBwdt
%
where A = x(ti) − x(ti−1) − f (x, t)d' and B = B(x, t)B(x, t)T . The transition
probability therefore is formulated as follows:
!
*[xi − %(ti; xi−1, ti−1]
#
=
"
dw
(2,)n exp
$
jwTA−
1
2
wTBwdt
%
(3.49)
This form of the transition probability is very common in the physics community. In
engineering fields, the transition probability is derived according to the distribution of
the state space noise that is considered to be Gaussian distributed. Therefore it seems
that the transition above is different than the one that would have been derived if we
were considering the Gaussian distribution of the state space noise. However as we will
show in the rest of our analysis, (3.49) takes the form of Gaussian distribution. More
precisely we will have that:
!
*[xi − %(ti; xi−1, ti−1]
#
=
"
dw
(2,)n exp
$
jwTBdt(Bdt)−1A−
1
2
wTBwdt
%
exp
$
−
j2
2 AT (Bdt)−TBdt(Bdt)−1A
%
93
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 107 |
| Contributing entity | University of Southern California |
| Repository email | cisadmin@lib.usc.edu |
| Full text | By substituting back we will have: ! *[xi − %(ti; xi−1, ti−1] # = " dw (2,)n exp $ jwT $ x(ti) − x(ti−1) − f (x, t)d' %% × exp $ − 1 2 wTB(x, t)B(x, t)Twdt % = " dw (2,)n exp $ jwTA % exp $ − 1 2 wTBwdt % where A = x(ti) − x(ti−1) − f (x, t)d' and B = B(x, t)B(x, t)T . The transition probability therefore is formulated as follows: ! *[xi − %(ti; xi−1, ti−1] # = " dw (2,)n exp $ jwTA− 1 2 wTBwdt % (3.49) This form of the transition probability is very common in the physics community. In engineering fields, the transition probability is derived according to the distribution of the state space noise that is considered to be Gaussian distributed. Therefore it seems that the transition above is different than the one that would have been derived if we were considering the Gaussian distribution of the state space noise. However as we will show in the rest of our analysis, (3.49) takes the form of Gaussian distribution. More precisely we will have that: ! *[xi − %(ti; xi−1, ti−1] # = " dw (2,)n exp $ jwTBdt(Bdt)−1A− 1 2 wTBwdt % exp $ − j2 2 AT (Bdt)−TBdt(Bdt)−1A % 93 |
