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the position and velocity of the trajectory, while zt, xt are internal states. 2z, )z, ' are
time constants. The nonlinear coupling or forcing term f is defined as:
f(x) =
CNi
=1 K (xt, ci) "ixt CNi
=1 K (xt, ci)
(g − y0) = #P (x)T ) (6.2)
The basis functions K (xt, ci) are defined as:
K (xt, ci) = wi = exp
A
−0.5hi(xt − ci)2B
(6.3)
with bandwidth hi and center ci of the Gaussian kernels – for more details see (Ijspeert
et al. 2003). The full dynamics have the form of dx = F(x)dt+G(x)udt where the state
x is specified as x = (x, y, z) while the controls are specified as u = ) = ("1, ..., "p)T .The
representation above is advantageous as it guarantees attractor properties towards the
goal while remaining linear in the parameters ) of the function approximator. By varying
the parameter ) the shape of the trajectory changes while the goal state g and initial
state yt0 remain fixed. These properties facilitate learning (Peters & Schaal 2008c).
6.1.2 Nonlinear limit cycle attractors with adjustable land-scape
The canonical system for the case of limit cycle attractors consist the differential equation
' ˙&
= 1 where the term & ! [0, 2,] correspond to the phase angle of the oscillator in polar
coordinates. The amplitude of the oscillation is assumed to be r. This oscillator produces
a stable limit cycle when projected into Cartesian coordinated with v1 = r cos(&) and
v2 = r sin(&). In fact, it corresponds to form of the (Hopf-like) oscillator equations
182
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 196 |
| Contributing entity | University of Southern California |
| Repository email | cisadmin@lib.usc.edu |
| Full text | the position and velocity of the trajectory, while zt, xt are internal states. 2z, )z, ' are time constants. The nonlinear coupling or forcing term f is defined as: f(x) = CNi =1 K (xt, ci) "ixt CNi =1 K (xt, ci) (g − y0) = #P (x)T ) (6.2) The basis functions K (xt, ci) are defined as: K (xt, ci) = wi = exp A −0.5hi(xt − ci)2B (6.3) with bandwidth hi and center ci of the Gaussian kernels – for more details see (Ijspeert et al. 2003). The full dynamics have the form of dx = F(x)dt+G(x)udt where the state x is specified as x = (x, y, z) while the controls are specified as u = ) = ("1, ..., "p)T .The representation above is advantageous as it guarantees attractor properties towards the goal while remaining linear in the parameters ) of the function approximator. By varying the parameter ) the shape of the trajectory changes while the goal state g and initial state yt0 remain fixed. These properties facilitate learning (Peters & Schaal 2008c). 6.1.2 Nonlinear limit cycle attractors with adjustable land-scape The canonical system for the case of limit cycle attractors consist the differential equation ' ˙& = 1 where the term & ! [0, 2,] correspond to the phase angle of the oscillator in polar coordinates. The amplitude of the oscillation is assumed to be r. This oscillator produces a stable limit cycle when projected into Cartesian coordinated with v1 = r cos(&) and v2 = r sin(&). In fact, it corresponds to form of the (Hopf-like) oscillator equations 182 |
