Convergence Rate for the Ordered Upwind Method

Shum, Alex; Morris, Kirsten; Khajepour, Amir

doi:10.1007/s10915-016-0163-3

Cited by 7 publications

(5 citation statements)

References 22 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Therefore, we formulate the path planning problem as finding the optimal heading function u i ∈ U (x 0 , x g ) that entails the minimal value of T (x 0 , x g ). This is analogous to the approaches of many past research works [17], [23], [25] based on the ideas of Dynamic Programming, and results on (2).…”

Section: A Problem Statementsupporting

confidence: 56%

“…Some approaches consider cost directions parallel to the reference frame axes [21], [22], but for most unstructured scenarios this method is not suitable and hence produces sub-optimal results [17]. This can be overcome by using the Ordered Upwind Method (OUM), a numerical solver method that works using the static Hamilton-Jacobi-Bellman (HJB) equation [23]. In fact, the Eikonal is the particular isotropic case of this equation, which admits the use of cost functions that vary with direction, i.e.…”

Section: A Approaches To Optimal Path Planningmentioning

confidence: 99%

“…To numerically solve this problem, as introduced in Section I there is already a fast method capable of achieving this: the OUM. This algorithm produces a viscous solution of (3), in the sense it overcomes any discontinuity that may appear in the real solution of T (x) [23]. Furthermore, in order to preserve the uniqueness of ψ(x), it is mandatory to make the speed profile function F , shown in (4), convex [17], [23].…”

Section: A Problem Statementmentioning

confidence: 99%

See 2 more Smart Citations

Optimal Path Planning using CAMIS: a Continuous Anisotropic Model for Inclined Surfaces

Sánchez-Ibáñez,

Pérez-del-Pulgar,

Serón

et al. 2021

Preprint

View full text Add to dashboard Cite

The optimal traverse of irregular terrains made by ground mobile robots heavily depends on the adequacy of the cost models used to plan the path they follow. The criteria to define optimality may be based on minimizing energy consumption and/or preserving the robot stability. This entails the proper assessment of anisotropy to account for the robot driving on top of slopes with different directions. To fulfill this demand, this paper presents the Continuous Anisotropic Model for Inclined Surfaces, a cost model compatible with anisotropic path planners like the bi-directional Ordered Upwind Method. This model acknowledges how the orientation of the robot with respect to any slope determines its energetic cost, considering the action of gravity and terramechanic effects such as the slippage. Moreover, the proposed model can be tuned to define a trade-off between energy minimization and Roll angle reduction. The results from two simulation tests demonstrate how, to find the optimal path in scenarios containing slopes, in certain situations the use of this model can be more advantageous than relying on isotropic cost functions. Finally, the outcome of a field experiment involving a skid-steering robot that drives on top of a real slope is also discussed.

show abstract

Section: A Problem Statementsupporting

confidence: 56%

Section: A Approaches To Optimal Path Planningmentioning

confidence: 99%

Section: A Problem Statementmentioning

confidence: 99%

See 1 more Smart Citation

Optimal Path Planning using CAMIS: a Continuous Anisotropic Model for Inclined Surfaces

Sánchez-Ibáñez,

Pérez-del-Pulgar,

Serón

et al. 2021

Preprint

View full text Add to dashboard Cite

show abstract

“…There are particular situations in which FMM produces accurate results under a certain level of anisotropy, such as having a cost function formulated in such a way thatit varies mostly in the directions parallel to the reference axes [ 24 , 216 ]. This is the case depicted in Figure 11 b. Sethian and Vladimirsky [ 215 ] proposed the use of an algorithm called the Ordered Upwind Method (OUM) to deal with the static HJB equation, the convergence rate of which was demonstrated by Shum et al [ 217 ]. Its main drawback is the increase in computational cost it entails, proportional to the anisotropy existing in the scenario.…”

Section: Optimal-control-based Path Planning Algorithmsmentioning

confidence: 99%

Path Planning for Autonomous Mobile Robots: A Review

Sánchez‐Ibáñez

Pérez-del-Pulgar

García-Cerezo

2021

Sensors

174

View full text Add to dashboard Cite

Providing mobile robots with autonomous capabilities is advantageous. It allows one to dispense with the intervention of human operators, which may prove beneficial in economic and safety terms. Autonomy requires, in most cases, the use of path planners that enable the robot to deliberate about how to move from its location at one moment to another. Looking for the most appropriate path planning algorithm according to the requirements imposed by users can be challenging, given the overwhelming number of approaches that exist in the literature. Moreover, the past review works analyzed here cover only some of these approaches, missing important ones. For this reason, our paper aims to serve as a starting point for a clear and comprehensive overview of the research to date. It introduces a global classification of path planning algorithms, with a focus on those approaches used along with autonomous ground vehicles, but is also extendable to other robots moving on surfaces, such as autonomous boats. Moreover, the models used to represent the environment, together with the robot mobility and dynamics, are also addressed from the perspective of path planning. Each of the path planning categories presented in the classification is disclosed and analyzed, and a discussion about their applicability is added at the end.

show abstract

“…• The OUM [23] ensures that the quasipotential of the whole field can be obtained by discretizing the phase space into a mesh and calculating the quasipotential of the nodes point by point. However, the calculation is extremely time-consuming.…”

Section: Control Strategymentioning

confidence: 99%

Controlling mean exit time of stochastic dynamical systems based on quasipotential and machine learning

Yuan

She

2022

Preprint

View full text Add to dashboard Cite

The mean exit time escaping basin of attraction in the presence of white noise is of practical importance in various scientific fields. In this work, we propose a strategy to control mean exit time of general stochastic dynamical systems to achieve a desired value based on the quasipotential concept and machine learning. Specifically, we develop a neural network architecture to compute the global quasipotential function. Then we design a systematic iterated numerical algorithm to calculate the controller for a given mean exit time. Moreover, we identify the most probable path between metastable attractors with help of the effective Hamilton-Jacobi scheme and the trained neural network. Numercal experiments demonstrate that our control strategy is effective and sufficiently accurate.

show abstract

Convergence Rate for the Ordered Upwind Method

Cited by 7 publications

References 22 publications

Optimal Path Planning using CAMIS: a Continuous Anisotropic Model for Inclined Surfaces

Optimal Path Planning using CAMIS: a Continuous Anisotropic Model for Inclined Surfaces

Path Planning for Autonomous Mobile Robots: A Review

Controlling mean exit time of stochastic dynamical systems based on quasipotential and machine learning

Contact Info

Product

Resources

About