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derivative of ' with respect to time. To see that we just need to apply the transformation
'(x, t) = $(x, T −t) = $(x, ' ) and thus we will have that (t't = −(.$. . The backward
kolmogorov PDE is now transformed to a the forward PDE:
(.$. =
1
2
tr
$
('xx't)BBT
%
; in [0, T)×"n×1 (3.71)
The PDE above is the forward Kolmogorov PDE which corresponds to SDEs without
the drift term and only diffusion term. In the most general cases, the transformation
'(x, t) = $(x, T − t) = $(x, ' ) of the backward Kolmogorov PDE results in a forward
PDE which does nor always correspond to the forward Kolmogorov PDE. In fact, this
is true only in the case F(x, t) = 0, q(x) = 0 and f (x, t) = 0 ,x ! "n×1, t ! [0, T)
and constant diffusion matrix B. For the most general case the transformation '(x, t) =
$(x, T − t) = $(x, ' ) results in the PDEs given by the equation that follows:
(.$(i)
. = −
1
%
q(x, T − ' )$(i)
. + f (i)
.
T ('x$(i)
. ) +
1
2
tr
?
('xx$(i)
. )BBT
@
+ F(x, T − ' )
(3.72)
with the initial condition $(x, 0) = exp(−1+
&(tN)). By substituting ˜q(x, ') = q(x, T −
' ) and ˜F
(x, ') = F(x, T − ' ) the Feynman Kac representation takes the form:
$(x, ') =
!
.(x0) exp
$
−
1
%
" .
0
˜q(x, s)ds
%
+
" .
0
˜F
(x, ") exp
$
−
1
%
" .
t
˜q(x, s)ds
%
d"
#
106
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 120 |
| Contributing entity | University of Southern California |
| Repository email | cisadmin@lib.usc.edu |
| Full text | derivative of ' with respect to time. To see that we just need to apply the transformation '(x, t) = $(x, T −t) = $(x, ' ) and thus we will have that (t't = −(.$. . The backward kolmogorov PDE is now transformed to a the forward PDE: (.$. = 1 2 tr $ ('xx't)BBT % ; in [0, T)×"n×1 (3.71) The PDE above is the forward Kolmogorov PDE which corresponds to SDEs without the drift term and only diffusion term. In the most general cases, the transformation '(x, t) = $(x, T − t) = $(x, ' ) of the backward Kolmogorov PDE results in a forward PDE which does nor always correspond to the forward Kolmogorov PDE. In fact, this is true only in the case F(x, t) = 0, q(x) = 0 and f (x, t) = 0 ,x ! "n×1, t ! [0, T) and constant diffusion matrix B. For the most general case the transformation '(x, t) = $(x, T − t) = $(x, ' ) results in the PDEs given by the equation that follows: (.$(i) . = − 1 % q(x, T − ' )$(i) . + f (i) . T ('x$(i) . ) + 1 2 tr ? ('xx$(i) . )BBT @ + F(x, T − ' ) (3.72) with the initial condition $(x, 0) = exp(−1+ &(tN)). By substituting ˜q(x, ') = q(x, T − ' ) and ˜F (x, ') = F(x, T − ' ) the Feynman Kac representation takes the form: $(x, ') = ! .(x0) exp $ − 1 % " . 0 ˜q(x, s)ds % + " . 0 ˜F (x, ") exp $ − 1 % " . t ˜q(x, s)ds % d" # 106 |
