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and only of the later admits an optimum value function with same features. Furthermore
the optimal control and value functions are identical and they are specified by:
u"(x) = −R−1GT'xV (2.119)
where
−(tVt = q + ('xV )T f −
1
2
('xV )TGR−1GT ('xV )
+
1
20
('xV ) ˜C!!˜C
T ('xV ) +
$
20
tr
?
'xx'˜C!!˜C
T
@
with boundary condition VtN = &tN , iff the following conditions holds ˜C
(x)!!˜C
(x)T =
C(x)!!C(x)T . The parameters 0, % > 0 and !! defined as !! = LLT .
The use of the two parameters $ and 0 in the analysis above may seem a bit confusing.
As a first observation, $ and 0 are tuning parameters in the cost function and therefore
it does not make sense to multiply the process noise in the stochastic dynamics since in
most control applications the stochastic dynamics are given and their uncertainty is not
a matter of manipulation and user tuning.
To resolve the confusion we consider the special case where $ = 0 which is the most
studied in the control literature. In this case the parameters $, 0 drop from the stochastic
dynamics and they appear only in the cost functions. When $ )= 0 this is a generalization
since we can now ask an additional question: Given the cost functions in risk sensitive and
differential game optimal control problems and the difference between the risk parameter
$ and the distrurbance weight 0 what is the form of the stochastic dynamics for which
54
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 68 |
| Contributing entity | University of Southern California |
| Repository email | cisadmin@lib.usc.edu |
| Full text | and only of the later admits an optimum value function with same features. Furthermore the optimal control and value functions are identical and they are specified by: u"(x) = −R−1GT'xV (2.119) where −(tVt = q + ('xV )T f − 1 2 ('xV )TGR−1GT ('xV ) + 1 20 ('xV ) ˜C!!˜C T ('xV ) + $ 20 tr ? 'xx'˜C!!˜C T @ with boundary condition VtN = &tN , iff the following conditions holds ˜C (x)!!˜C (x)T = C(x)!!C(x)T . The parameters 0, % > 0 and !! defined as !! = LLT . The use of the two parameters $ and 0 in the analysis above may seem a bit confusing. As a first observation, $ and 0 are tuning parameters in the cost function and therefore it does not make sense to multiply the process noise in the stochastic dynamics since in most control applications the stochastic dynamics are given and their uncertainty is not a matter of manipulation and user tuning. To resolve the confusion we consider the special case where $ = 0 which is the most studied in the control literature. In this case the parameters $, 0 drop from the stochastic dynamics and they appear only in the cost functions. When $ )= 0 this is a generalization since we can now ask an additional question: Given the cost functions in risk sensitive and differential game optimal control problems and the difference between the risk parameter $ and the distrurbance weight 0 what is the form of the stochastic dynamics for which 54 |
