Rates and mean first passage times

Müller, Reinhard; Talkner, Peter; Reimann, Peter

doi:10.1016/s0378-4371(97)00390-7

Cited by 18 publications

(18 citation statements)

References 23 publications

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“…Such systems cannot be classified as "stable", but it is also misleading and incomplete to classify them as "unstable". Physicists have long used the term metastable to capture this interesting phenomenon and have developed a number of tools for quantifying this behavior (Hanggi, Talkner, & Borkovec, 1990;Kampen, 2007;Muller, Talkner, & Reimann, 1997;Talkner, Hanggi, Freidkin, & Trautmann, 1987). Many other branches of science and engineering have also borrowed this terminology to describe dynamic systems in a wide variety of fields.…”

Section: Introductionmentioning

confidence: 99%

Metastable Walking Machines

Byl

Tedrake

2009

The International Journal of Robotics Research

153

143

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Abstract-Legged robots that operate in the real world are inherently subject to stochasticity in their dynamics and uncertainty about the terrain. Due to limited energy budgets and limited control authority, these "disturbances" cannot always be canceled out with high-gain feedback. Minimally-actuated walking machines subject to stochastic disturbances no longer satisfy strict conditions for limit-cycle stability; however, they can still demonstrate impressively long-living periods of continuous walking. Here, we employ tools from stochastic processes to examine the "stochastic stability" of idealized rimless-wheel and compass-gait walking on randomly generated uneven terrain. Furthermore, we employ tools from numerical stochastic optimal control to design a controller for an actuated compass gait model which maximizes a measure of stochastic stability -the mean first-passage-time -and compare its performance to a deterministic counterpart. Our results demonstrate that walking is well-characterized as a metastable process, and that the stochastic dynamics of walking should be accounted for during control design in order to improve the stability of our machines.

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Section: Introductionmentioning

confidence: 99%

Metastable Walking Machines

Byl

Tedrake

2009

The International Journal of Robotics Research

153

143

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“…Such systems cannot be classified as "stable", but it is also misleading and incomplete to classify them as "unstable". Physicists have long used the term metastable to capture this interesting phenomenon and have developed a number of tools for quantifying this behavior [8,9,12,15]. Many other branches of science and engineering have also borrowed the terminology to describe dynamic systems in a wide variety of fields.…”

Section: Introductionmentioning

confidence: 99%

Metastable Walking on Stochastically Rough Terrain

Byl

Tedrake

2008

Robotics: Science and Systems IV

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Abstract-Simplified models of limit-cycle walking on flat terrain have provided important insights into the nature of legged locomotion. Real walking robots (and humans), however, do not exhibit true limit cycle dynamics because terrain, even in a carefully designed laboratory setting, is inevitably non-flat. Walking systems on stochastically rough terrain may not satisfy strict conditions for limit-cycle stability but can still demonstrate impressively long-living periods of continuous walking. Here, we examine the dynamics of rimless-wheel and compass-gait walking on randomly generated rough terrain and employ tools from stochastic processes to describe the 'stochastic stability' of these gaits. This analysis generalizes our understanding of walking stability and may provide statistical tools for experimental limit cycle analysis on real walking systems.

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“…the average time that a random walker starting out from a point x 0 inside the initial domain of attraction, assumes in order to leave the attracting domain for the first time [2,3,4]. Put differently, the MFPT is the average time needed to cross the deterministic separatrix-manifold for the first time [2,6,7]. At weak noise the MFPT becomes essentially independent of the starting point, i.e.…”

Section: Introductionmentioning

confidence: 99%

“…In particular, the MFPT analysis requires the choice of a correct boundary condition [6,7]. These are well known for one-dimensional stochastic diffusion Markov processes x(t) which are of the FokkerPlanck form, Eqs.…”

Section: Introductionmentioning

confidence: 99%

Escape driven byα-stable white noises

2007

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We explore the archetype problem of an escape dynamics occurring in a symmetric double well potential when the Brownian particle is driven by white Lévy noise in a dynamical regime where inertial effects can safely be neglected. The behavior of escaping trajectories from one well to another is investigated by pointing to the special character that underpins the noise-induced discontinuity which is caused by the generalized Brownian paths that jump beyond the barrier location without actually hitting it. This fact implies that the boundary conditions for the mean first passage time (MFPT) are no longer determined by the well-known local boundary conditions that characterize the case with normal diffusion. By numerically implementing properly the set up boundary conditions, we investigate the survival probability and the average escape time as a function of the corresponding Lévy white noise parameters. Depending on the value of the skewness β of the Lévy noise, the escape can either become enhanced or suppressed: a negative asymmetry β causes typically a decrease for the escape rate while the rate itself depicts a non-monotonic behavior as a function of the stability index α which characterizes the jump length distribution of Lévy noise, with a marked discontinuity occurring at α = 1. We find that the typical factor of "two" that characterizes for normal diffusion the ratio between the MFPT for well-bottom-to-well-bottom and well-bottom-tobarrier-top no longer holds true. For sufficiently high barriers the survival probabilities assume an exponential behavior. Distinct non-exponential deviations occur, however, for low barrier heights.

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Rates and mean first passage times

Cited by 18 publications

References 23 publications

Metastable Walking Machines

Metastable Walking Machines

Metastable Walking on Stochastically Rough Terrain

Escape driven byα-stable white noises

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