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The term exp
A
−+Nl
2 log (2,dt%)
B
does not depend on the trajectory " i, therefore it
can be taken outside the integral as well as outside the gradient. Thus we will have that:
uti = lim
dt&0
8
:%R−1GTt
i
exp
?
−+(N−i)l
2 log (2,dt%)
@
'xti
6F
exp
?
−1+
˜ S(" i)
@
d" i
7
exp
?
−+(N−i)l
2 log (2,dt%)
@ F
exp
?
−1+
˜ S(" i)
@
d" i
;
=.
The constant term drops from the nominator and denominator and thus we can write:
uti = lim
dt&0
8
:%R−1GTt
i
0
2'xti
F
exp
?
−1+
˜ S(" i)
@
d" i
F
exp
?
−1+
˜ S(" i)
@
d" i
3
5
;
=.
Under the assumption that term exp
?
−1+
˜ S(" i)
@
d" i is continuously differentiable in
xti and dt we can change order of the integral with the differentiation operations. In
general for 'x
F
f(x, y)dy =
F
'xf(x, y)dy to be true, f(x, t) should be continuous in
y and differentiable in x. Under this assumption, the optimal controls can be further
formulated as:
uti = lim
dt&0
0
2%R−1GTt
i
F
'xti exp
?
−1+
˜ S(" i)
@
d" i
F
exp
?
−1+
˜ S(" i)
@
d" i
3
5 .
Application of the differentiation rule of the exponent results in:
uti = lim
dt&0
0
2%R−1GTt
i
F
exp
?
−1+
˜ S(" i)
@
'xti
?
−1+
˜ S(" i)
@
d" i
F
exp
?
−1+
˜ S(" i)
@
d" i
3
5 .
The denominator is a function of xti the current state and thus it can be pushed inside
the integral of the nominator:
154
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 168 |
| Contributing entity | University of Southern California |
| Repository email | cisadmin@lib.usc.edu |
| Full text | The term exp A −+Nl 2 log (2,dt%) B does not depend on the trajectory " i, therefore it can be taken outside the integral as well as outside the gradient. Thus we will have that: uti = lim dt&0 8 :%R−1GTt i exp ? −+(N−i)l 2 log (2,dt%) @ 'xti 6F exp ? −1+ ˜ S(" i) @ d" i 7 exp ? −+(N−i)l 2 log (2,dt%) @ F exp ? −1+ ˜ S(" i) @ d" i ; =. The constant term drops from the nominator and denominator and thus we can write: uti = lim dt&0 8 :%R−1GTt i 0 2'xti F exp ? −1+ ˜ S(" i) @ d" i F exp ? −1+ ˜ S(" i) @ d" i 3 5 ; =. Under the assumption that term exp ? −1+ ˜ S(" i) @ d" i is continuously differentiable in xti and dt we can change order of the integral with the differentiation operations. In general for 'x F f(x, y)dy = F 'xf(x, y)dy to be true, f(x, t) should be continuous in y and differentiable in x. Under this assumption, the optimal controls can be further formulated as: uti = lim dt&0 0 2%R−1GTt i F 'xti exp ? −1+ ˜ S(" i) @ d" i F exp ? −1+ ˜ S(" i) @ d" i 3 5 . Application of the differentiation rule of the exponent results in: uti = lim dt&0 0 2%R−1GTt i F exp ? −1+ ˜ S(" i) @ 'xti ? −1+ ˜ S(" i) @ d" i F exp ? −1+ ˜ S(" i) @ d" i 3 5 . The denominator is a function of xti the current state and thus it can be pushed inside the integral of the nominator: 154 |
