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• Given:
– The system dynamics xti+1 = xti + (fti +Gtut) dt + Btidwti (cf. 4.2)
– The immediate cost Lt = qt + 1
2uTt
Rut (cf. 4.3)
– A terminal cost term &tN
– Trajectory starting at ti and ending at tN: " i = (xti , ....., xtN )
– A partitioning of the system dynamics into (c) controlled and (m) uncontrolled
equations, where n = c + m is the dimensionality of the state xt (cf. Section
4.2)
• Optimal Controls:
– Optimal controls at every time step ti: utidt =
F
P (" i) uL (" i) d" (c)
i
– Probability of a trajectory: P (" i) = e−1"
˜ S(" i)
!
e−1"
˜ S(" i)d" i
– Generalized trajectory cost: ˜ S(" i) = S(" i) + +2
CN−1
j=i log |Btj | where
& S(" i) = &tN +
CN−1
j=i qtj dt + 1
2
CN−1
j=i
MMMMM
x(c)
tj+1−x(c)
tj
dt − f (c)
tj
MMMMM
2
H−1
tj
dt
& Htj = G(c)
tj R−1G(c)
tj
T
and B = B(c)
tj B(c)
tj
T
– Local Controls: uL(" i) = R−1G(c)
ti
T
?
G(c)
ti R−1G(c)
ti
T
@
−1 ?
G(c)
ti dwti
@
.
Table 4.1: Summary of optimal control derived from the path integral formalizm.
Given that this result is of general value and constitutes the foundation to derive our
reinforcement learning algorithm in the next section, but also since many other special
cases can be derived from it, we summarized all relevant equations in Table 4.1.
The Given components of Table 4.1 include a model of the system dynamics, the
cost function, knowledge of the system’s noise process, and a mechanism to generate
trajectories " i. It is important to realize that this is a model-based approach, as the
computations of the optimal controls requires knowledge of !i. !i can be obtained in two
ways. First, the trajectories " i can be generated purely in simulation, where the noise
133
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 147 |
| Contributing entity | University of Southern California |
| Repository email | cisadmin@lib.usc.edu |
| Full text | • Given: – The system dynamics xti+1 = xti + (fti +Gtut) dt + Btidwti (cf. 4.2) – The immediate cost Lt = qt + 1 2uTt Rut (cf. 4.3) – A terminal cost term &tN – Trajectory starting at ti and ending at tN: " i = (xti , ....., xtN ) – A partitioning of the system dynamics into (c) controlled and (m) uncontrolled equations, where n = c + m is the dimensionality of the state xt (cf. Section 4.2) • Optimal Controls: – Optimal controls at every time step ti: utidt = F P (" i) uL (" i) d" (c) i – Probability of a trajectory: P (" i) = e−1" ˜ S(" i) ! e−1" ˜ S(" i)d" i – Generalized trajectory cost: ˜ S(" i) = S(" i) + +2 CN−1 j=i log |Btj | where & S(" i) = &tN + CN−1 j=i qtj dt + 1 2 CN−1 j=i MMMMM x(c) tj+1−x(c) tj dt − f (c) tj MMMMM 2 H−1 tj dt & Htj = G(c) tj R−1G(c) tj T and B = B(c) tj B(c) tj T – Local Controls: uL(" i) = R−1G(c) ti T ? G(c) ti R−1G(c) ti T @ −1 ? G(c) ti dwti @ . Table 4.1: Summary of optimal control derived from the path integral formalizm. Given that this result is of general value and constitutes the foundation to derive our reinforcement learning algorithm in the next section, but also since many other special cases can be derived from it, we summarized all relevant equations in Table 4.1. The Given components of Table 4.1 include a model of the system dynamics, the cost function, knowledge of the system’s noise process, and a mechanism to generate trajectories " i. It is important to realize that this is a model-based approach, as the computations of the optimal controls requires knowledge of !i. !i can be obtained in two ways. First, the trajectories " i can be generated purely in simulation, where the noise 133 |
