First-excursion times have been developed extensively in the literature for oscillators; one major application is structural dynamics of buildings. Using the fact that most closed-loop systems operate with a moderate to high damping ratio, we have derived a new procedure for calculating first-excursion times for a class of linear continuous, time-varying systems. In several examples, we show that the algorithm is both accurate and time-efficient. These are important attributes for real-time path planning in stochastic environments, and hence the work should be useful for autonomous robotic systems involving marine and air vehicles.
References
1.
Harris
, S.
, 1981, “Steady, One-Dimensional Brownian Motion With an Absorbing Boundary
,” J. Chem. Phys.
, 75
(6
), pp. 3103
–3106
.2.
Montroll
, E. W.
, and Scher
, H.
, 1973, “Random Walks on Lattices: Continuous-time Walks and Influence of Absorbing Boundaries
,” J. Stat. Phys.
, 9
(2
), pp. 101
–135
.3.
Bach
, E.
, Coppersmith
, S.
, Goldschen
, M. P.
, Joynt
, R.
, and Watrous
, J.
, 2004, “One-Dimensional Quantum Walks With Absorbing Boundaries
,” J. Comput. Syst. Sci.
, 69
(4
), pp. 562
–592
.4.
Burkhardt
, T. W.
, 2002, “Absorption of a Randomly Accelerated Particle: Recent Results for Partially Absorbing and Inelastic Boundaries
,” Physica A
, 306
, pp. 107
–116
.5.
Crandall
, S.
, 1970, “First-Crossing Probabilities of the Linear Oscillator
,” J. Sound Vib.
, 12
(3
), pp. 285
–299
.6.
Vanmarcke
, E.
, 1975, “Distribution of First-Passage Time for Normal Stationary Random Processes
,” ASME J. Appl. Mech.
, 42
(1
), pp. 215
–220
.7.
Langley
, R.
, 1988, “A First Passage Approximation for Normal Stationary Random Processes
,” J. Sound Vib.
, 122
(2
), pp. 261
–275
.8.
Lin
, Y.
, and Cai
, G.
, Probabilistic Structural Mechanics
(McGraw-Hill
, New York
, 2004).9.
He
, J.
, 2009, “Numerical Calculation for First Excursion Probabilities for Linear Systems
,” Probab. Eng. Mech.
, 24
, pp. 418
–425
.10.
Au
, S.
, and Beck
, J.
, 2001, “First Excursion Probabilities for Linear Systems by Very Efficient Importance Sampling
,” Probab. Eng. Mech.
, 16
, pp. 193
–207
.11.
Erdmann
, M.
, 1985, “Using Backprojections for Fine Motion Planning With Uncertainty
,” Proceedings of the 1985 IEEE International Conference on Robotics and Automation
, Vol. 2
, pp. 549
–554
.12.
Blackmore
, L.
, 2006, “A Probabilistic Particle Control Approach to Optimal, Robust Predictive Control
,” Proceedings of the AIAA Guidance, Navigation and Control Conference
.13.
Roy
, N.
, and Thrun
, S.
, 1999, “Coastal Navigation With Mobile Robots
,” Adv. Neural Process. Syst.
, 12
, pp. 1043
–1049
.14.
Greytak
, M.
, and Hover
, F.
, 2009, “Motion Planning With An Analytic Risk Cost for Holonomic Vehicles
,” Proceedings of the 2009 IEEE Conference on Decision and Control
, pp. 5655
–5660
.15.
Hockenberry
, J.
, and Lesieutre
, B.
, 2004, “Evaluation of Uncertainty in Dynamic Simulations of Power System Models: The Probabilistic Collocation Method
,” IEEE Trans. Power Syst.
, 19
(3
), pp. 1483
–1491
.Copyright © 2011
by American Society of Mechanical Engineers
You do not currently have access to this content.
