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E4 =
!
*xTt
ATt
VxxOd
#
+
!
*uTt
BT
t VxxOd
#
+
!
OTd
VxxBt*ut
#
+
!
OTd
VxxAt*xt
#
E5 =
!
OTd
VxxOd
#
In the first category we have all these terms that depend neither on $t and nor on
Od(*x, *u, $t, *t). These are the terms that define E1. The second category E2 includes
terms that depend on $t but not on Od(*x, *u, $t, *t). In the third class E3, there are
terms that depends both on Od(*x, *u, $t, *t) and $t . In the fourth class E4, we have
terms that depend on Od(*x, *u, $t, *t). Finally in the fifth class E5, we have all these
terms that depend on Od(*x, *u, $t, *t) quadratically. The expectation operator will
cancel all the terms that include noise up the first order. Moreover, the mean operator
for terms that depend on the noise quadratically will result in covariance.
We compute the expectations of all the terms in the E1 class. More precisely we will
have that:
!
*xTt
ATt
VxxAt*xt
#
= *xTt
ATt
VxxAt*xt
!
*uTt
BT
t VxxBt*ut
#
= *uTt
BT
t VxxBt*ut
!
*xTt
ATt
VxxBt*ut
#
= *xTt
ATt
VxxBt*ut
!
*uTt
BT
t VxxAt*xt
#
= *uTt
BT
t VxxAt*xt
(2.48)
We continue our analysis by calculating all the terms in the class E2. More presicely
we will have:
32
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 46 |
| Contributing entity | University of Southern California |
| Repository email | cisadmin@lib.usc.edu |
| Full text | E4 = ! *xTt ATt VxxOd # + ! *uTt BT t VxxOd # + ! OTd VxxBt*ut # + ! OTd VxxAt*xt # E5 = ! OTd VxxOd # In the first category we have all these terms that depend neither on $t and nor on Od(*x, *u, $t, *t). These are the terms that define E1. The second category E2 includes terms that depend on $t but not on Od(*x, *u, $t, *t). In the third class E3, there are terms that depends both on Od(*x, *u, $t, *t) and $t . In the fourth class E4, we have terms that depend on Od(*x, *u, $t, *t). Finally in the fifth class E5, we have all these terms that depend on Od(*x, *u, $t, *t) quadratically. The expectation operator will cancel all the terms that include noise up the first order. Moreover, the mean operator for terms that depend on the noise quadratically will result in covariance. We compute the expectations of all the terms in the E1 class. More precisely we will have that: ! *xTt ATt VxxAt*xt # = *xTt ATt VxxAt*xt ! *uTt BT t VxxBt*ut # = *uTt BT t VxxBt*ut ! *xTt ATt VxxBt*ut # = *xTt ATt VxxBt*ut ! *uTt BT t VxxAt*xt # = *uTt BT t VxxAt*xt (2.48) We continue our analysis by calculating all the terms in the class E2. More presicely we will have: 32 |
