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'v˙1 = −μ
L
v2
1 + v2
L 2 − r
v2
1 + v2
2
v1 − v2 (6.4)
'v˙2 = −μ
L
v2
1 + v2
L 2 − r
v2
1 + v2
2
v2 + v1 (6.5)
where μ is a positive time constant. The system above evolve to the limit cycle
v1 = r cos(t/' + c) and v2 = r sin(t/' + c) with c a constant, given any initial conditions
except [v1, v2] = [0, 0] which is an unstable fixed point. Therefore the canonical system
provides the amplitude signal (r) and a phase signal (&) to the forcing term:
f(&, r) =
CNi
=1 K (&, ci) "i CNi
=1 K (&, ci)
r = #R(&)T ) (6.6)
where the basis function K (&, ci) are defined as K (&, ci) = exp (hi (cos(& − ci) − 1)).
The forcing term is incorporated into the transformation system which is expressed by
the equations (6.1). The full dynamics of the rhythmic movement primitives have the
form of dx = F(x)dt+G(x)udt where the state x is specified as x = (&, v1, v2, z, y) while
the controls are specified as u = ) = ("1, ..., "p)T . The term g for the case of limit cycle
attractors is interpreted as anchor point (or set point) for the oscillatory trajectory, which
can be changed to accommodate any desired baseline of the oscillation The complexity
of attractors is restricted only by the abilities of the function approximator used to gen-erate
the forcing term, which essentially allows for almost arbitrarily complex (smooth)
attractors with modern function approximators
183
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 197 |
| Contributing entity | University of Southern California |
| Repository email | cisadmin@lib.usc.edu |
| Full text | 'v˙1 = −μ L v2 1 + v2 L 2 − r v2 1 + v2 2 v1 − v2 (6.4) 'v˙2 = −μ L v2 1 + v2 L 2 − r v2 1 + v2 2 v2 + v1 (6.5) where μ is a positive time constant. The system above evolve to the limit cycle v1 = r cos(t/' + c) and v2 = r sin(t/' + c) with c a constant, given any initial conditions except [v1, v2] = [0, 0] which is an unstable fixed point. Therefore the canonical system provides the amplitude signal (r) and a phase signal (&) to the forcing term: f(&, r) = CNi =1 K (&, ci) "i CNi =1 K (&, ci) r = #R(&)T ) (6.6) where the basis function K (&, ci) are defined as K (&, ci) = exp (hi (cos(& − ci) − 1)). The forcing term is incorporated into the transformation system which is expressed by the equations (6.1). The full dynamics of the rhythmic movement primitives have the form of dx = F(x)dt+G(x)udt where the state x is specified as x = (&, v1, v2, z, y) while the controls are specified as u = ) = ("1, ..., "p)T . The term g for the case of limit cycle attractors is interpreted as anchor point (or set point) for the oscillatory trajectory, which can be changed to accommodate any desired baseline of the oscillation The complexity of attractors is restricted only by the abilities of the function approximator used to gen-erate the forcing term, which essentially allows for almost arbitrarily complex (smooth) attractors with modern function approximators 183 |
