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subject to the stochastic dynamics: dx = (f (x) + G(x)u)dt + B(x)d-. We define the
differential entropy:
H
$
u(x, t), p(u), x(t0)
%
= −
"
"x0
"
"x
p
$
u, x(t0)
%
log p
$
u, x(t0)
%
dx dx0 (2.121)
where p
$
u, x(t0)
%
is the probability of selecting u while x0 is the initial state and
(x,(x0 the spaces of the states and the initial conditions. Next, we are looking for the
probability distribution which best represents the random variable u. The answer to this
request is given by Jayne’s maximum entropy principle which states that the best distribu-tion
is the one that maximizes the entropy formulation above. This maximization proce-dure
is subjected to the constrains that E
$
J(u, x)
%
= K and also
F
p
$
u, x(t0)
%
dx0 = 1.
As stated in (Saridis 1996), this problem is more general than the optimal control since
the parameter K is fixed and unknown and it depends on the selection of the controls
u(x, t). The unconstrained maximization problem is now formulated as follows:
) = ) H
$
u(x, t), p(u), x(t0)
%
− 0
$
E
$
J(u, x)
%
− K
%
− 2
$"
p
$
u, x(t0)
%
dx0 − 1
%
/ −
" $
) p
$
u, x(t0)
%
log p
$
u, x(t0)
%
+0 p
$
u, x(t0)
%
J(u, x)
%
dx
− 2
$"
p(u, x(t0))dx0 − 1
%
(2.122)
The objective function above is concave with respect to the probability distribution
since the second derivative !#
!p = −)1p
< 0. Thus to find the maximum we take the first
56
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 70 |
| Contributing entity | University of Southern California |
| Repository email | cisadmin@lib.usc.edu |
| Full text | subject to the stochastic dynamics: dx = (f (x) + G(x)u)dt + B(x)d-. We define the differential entropy: H $ u(x, t), p(u), x(t0) % = − " "x0 " "x p $ u, x(t0) % log p $ u, x(t0) % dx dx0 (2.121) where p $ u, x(t0) % is the probability of selecting u while x0 is the initial state and (x,(x0 the spaces of the states and the initial conditions. Next, we are looking for the probability distribution which best represents the random variable u. The answer to this request is given by Jayne’s maximum entropy principle which states that the best distribu-tion is the one that maximizes the entropy formulation above. This maximization proce-dure is subjected to the constrains that E $ J(u, x) % = K and also F p $ u, x(t0) % dx0 = 1. As stated in (Saridis 1996), this problem is more general than the optimal control since the parameter K is fixed and unknown and it depends on the selection of the controls u(x, t). The unconstrained maximization problem is now formulated as follows: ) = ) H $ u(x, t), p(u), x(t0) % − 0 $ E $ J(u, x) % − K % − 2 $" p $ u, x(t0) % dx0 − 1 % / − " $ ) p $ u, x(t0) % log p $ u, x(t0) % +0 p $ u, x(t0) % J(u, x) % dx − 2 $" p(u, x(t0))dx0 − 1 % (2.122) The objective function above is concave with respect to the probability distribution since the second derivative !# !p = −)1p < 0. Thus to find the maximum we take the first 56 |
