First Passage Time for Brownian Motion and Piecewise Linear Boundaries

Jin, Zhigang; Wang, Liqun

doi:10.1007/s11009-015-9475-2

Cited by 10 publications

(5 citation statements)

References 17 publications

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“…This paper leans heavily on the classical notion of firstpassage times, a well-studied topic in the field of continuous stochastic processes. Indeed, the piecewise approximation presented in Section IV of this paper is similar in spirit to that of Jin and Wang [10], although we consider the case where the process (rather than purely the boundary) is nonlinear.…”

Section: Stochastic Processes In Continuous-timementioning

confidence: 99%

“…It is straightforward to verify that the total exit probability can be written as F(T ) = P (τ 0 ≤ T ), and thus τ 0 offers a means by which to analyze the time-evolution of F(t). In some cases F(t) is known to be time-differentiable [10], and thus computing this derivative (called the first-passage density) would seem to be a natural goal. However, this computation is challenging for generic nonlinear processes, and instead we will settle for an interval-based "integration" scheme that reflects how F(T ) can be approximated in practice.…”

Section: How Risk Evolves In Timementioning

confidence: 99%

See 1 more Smart Citation

Collision Probabilities for Continuous-Time Systems Without Sampling

Frey¹,

Steiner²,

How³

2020

Robotics: Science and Systems XVI

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Demand for high-performance, robust, and safe autonomous systems has grown substantially in recent years. These objectives motivate the desire for efficient risk estimation that can be embedded in core decision-making tasks such as motion planning. On one hand, Monte-Carlo (MC) and other samplingbased techniques provide accurate solutions for a wide variety of motion models but are cumbersome in the context of continuous optimization. On the other hand, "direct" approximations aim to compute (or upper-bound) the failure probability as a smooth function of the decision variables, and thus are convenient for optimization. However, existing direct approaches fundamentally assume discrete-time dynamics and can perform unpredictably when applied to continuous-time systems ubiquitous in the real world, often manifesting as severe conservatism. State-of-theart attempts to address this within a conventional discretetime framework require additional Gaussianity approximations that ultimately produce inconsistency of their own. In this paper we take a fundamentally different approach, deriving a risk approximation framework directly in continuous time and producing a lightweight estimate that actually converges as the underlying discretization is refined. Our approximation is shown to significantly outperform state-of-the-art techniques in replicating the MC estimate while maintaining the functional and computational benefits of a direct method. This enables robust, risk-aware, continuous motion-planning for a broad class of nonlinear, partially-observable systems.

show abstract

Section: Stochastic Processes In Continuous-timementioning

confidence: 99%

Section: How Risk Evolves In Timementioning

confidence: 99%

Collision Probabilities for Continuous-Time Systems Without Sampling

Frey¹,

Steiner²,

How³

2020

Robotics: Science and Systems XVI

View full text Add to dashboard Cite

show abstract

“…This paper leans heavily on the classical notion of firstpassage times, a well-studied topic in the field of continuous stochastic processes. Indeed, the piecewise approximation presented in Section IV of this paper is similar in spirit to that of Jin and Wang [9], although we consider the case where the process (rather than purely the boundary) is nonlinear.…”

Section: Stochastic Processes In Continuous-timementioning

confidence: 99%

“…It is straightforward to verify that the total exit probability can be written as F(T ) = P (τ 0 ≤ T ), and thus τ 0 offers a means by which to analyze the time-evolution of F(t). In some cases F(t) is known to be time-differentiable [9], and thus computing this derivative (called the first-passage density) would seem to be a natural goal. However, this computation is challenging for generic nonlinear processes, and instead we will settle for an interval-based "integration" scheme that reflects how F(T ) can be approximated in practice.…”

Section: How Risk Evolves In Timementioning

confidence: 99%

Collision Probabilities for Continuous-Time Systems Without Sampling [with Appendices]

Frey¹,

Steiner²,

How³

2020

Preprint

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Demand for high-performance, robust, and safe autonomous systems has grown substantially in recent years. Fulfillment of these objectives requires accurate and efficient risk estimation that can be embedded in core decision-making tasks such as motion planning. On one hand, Monte-Carlo (MC) and other sampling-based techniques can provide accurate solutions for a wide variety of motion models but are cumbersome to apply in the context of continuous optimization. On the other hand, "direct" approximations aim to compute (or upper-bound) the failure probability as a smooth function of the decision variables, and thus are widely applicable. However, existing approaches fundamentally assume discrete-time dynamics and can perform unpredictably when applied to continuous-time systems operating in the real world, often manifesting as severe conservatism. Stateof-the-art attempts to address this within a conventional discretetime framework require additional Gaussianity approximations that ultimately produce inconsistency of their own. In this paper we take a fundamentally different approach, deriving a risk approximation framework directly in continuous time and producing a lightweight estimate that actually improves as the discretization is refined. Our approximation is shown to significantly outperform state-of-the-art techniques in replicating the MC estimate while maintaining the functional and computational benefits of a direct method. This enables robust, riskaware, continuous motion-planning for a broad class of nonlinear, partially-observable systems.

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“…In this subsection, the first passage probability of the BM through a piecewise linear boundary is used to estimate the RUL. The boundary crossing probability for BM is an important tool in stochastic modeling, and its efficiency has been proved in numerous research fields, such as epidemiology, environmental science, physics, as well as in mechanical engineering, [40], [41]. In what follows, the future environment is totally determined.…”

Section: A Rul Under Determined Future Environmentmentioning

confidence: 99%

Lubrication Oil Anti-Wear Property Degradation Modeling and Remaining Useful Life Estimation of the System Under Multiple Changes Operating Environment

2019

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Anti-wear property (AWP) is one of the critical characteristics that define the lubrication performance of the lubrication oil. The AWP deterioration is characterized by the wear rate variation and is often related to the operating environment. Since a low AWP can lead to system failure, proactive means to predict the remaining useful life (RUL) considering the environmental factors is an important practical relevance. This paper presents a stochastic model to determine the oil AWP deterioration in order to predict the RUL of the related system. The model assumes that the operating environment behaves as a continuous-time Markov chain (CTMC). A Bayesian methodology using three sources of information (online degradation information, observed degradation, and environmental data) is applied to update dynamically the RUL. In order to demonstrate the applicability of the proposed model, a case of study is presented. Furthermore, to show the accuracy and effectiveness of the proposed approach, a comparative study is conducted with a previously developed model, which does not consider the operating environment. INDEX TERMS Anti-wear property, Bayesian updating, lubrication oil, varying operating environment.

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First Passage Time for Brownian Motion and Piecewise Linear Boundaries

Cited by 10 publications

References 17 publications

Collision Probabilities for Continuous-Time Systems Without Sampling

Collision Probabilities for Continuous-Time Systems Without Sampling

Collision Probabilities for Continuous-Time Systems Without Sampling [with Appendices]

Lubrication Oil Anti-Wear Property Degradation Modeling and Remaining Useful Life Estimation of the System Under Multiple Changes Operating Environment

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