Path planning using Laplace's equation

Connolly, Christopher I.; Burns, J. Brian; Weiss, Richard

doi:10.1109/robot.1990.126315

Cited by 389 publications

(260 citation statements)

References 5 publications

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“…They are frequently used as vector fields where one can find the optimal velocity vector at any point in space [9], [17], [21]. In this work, we do not compute a vector field but rather find a locally optimal configuration for an active cannula given a target and a cost function.…”

Section: Related Workmentioning

confidence: 99%

Planning active cannula configurations through tubular anatomy

Lyons

Webster

Alterovitz

2010

2010 IEEE International Conference on Robotics and Automation

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Abstract-Medical procedures such as lung biopsy and brachytherapy require maneuvering through tubular structures such as the trachea and bronchi to reach clinical targets. We introduce a new method to plan configurations for active cannulas, medical devices composed of thin, pre-curved, telescoping lumens that are capable of following controlled, curved paths through open or liquid-filled cavities. Planning optimal configurations for these devices is challenging due to their complex kinematics, which involve both beam mechanics and space curves. In this paper, we propose an optimization-based planning algorithm that computes active cannula configurations through tubular structures that reach specified targets. Given the target location, the start position and orientation, and a geometric representation of the physical environment extracted from pre-procedure medical images, the planner optimizes insertion length and orientation angle of each lumen of the active cannula. The planner models active cannula kinematics using a physically-based simulation that incorporates beam mechanics and minimizes energy. The algorithm typically computes plans in less than 2 minutes on a standard PC. We apply the method in simulation to anatomy extracted from a human CT scan and demonstrate configurations for a 5-lumen active cannula that maneuver it through the bronchi to targets in the lung.

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Section: Related Workmentioning

confidence: 99%

Planning active cannula configurations through tubular anatomy

Lyons

Webster

Alterovitz

2010

2010 IEEE International Conference on Robotics and Automation

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“…In the harmonic function method, Laplace's equation is applied to the path planning problem [18]. Although the resulting potential field again does not have local minima, off-line computation is required to provide a solution to the Laplace equation.…”

Section: The Local Minimum Problemmentioning

confidence: 99%

Solving the potential field local minimum problem using internal agent states

Mabrouk

McInnes

2008

Robotics and Autonomous Systems

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We propose a new, extended artificial potential field method, which uses dynamic internal agent states. The internal states are modeled as a dynamical system of coupled first order differential equations that manipulate the potential field in which the agent is situated. The internal state dynamics are forced by the interaction of the agent with the external environment. Local equilibria in the potential field are then manipulated by the internal states and transformed from stable equilibria to unstable equilibria, allowing escape from local minima in the potential field. This new methodology successfully solves reactive path planning problems, such as a complex maze with multiple local minima, which cannot be solved using conventional static potential fields.

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“…Connolly et al [4] and Akishita et al [5] independently developed a global method using solutions to Laplace's equations for path planning to generate a smooth, collision-free path. The potential field is computed in a global manner, i.e.…”

Section: Introductionmentioning

confidence: 99%

“…performing the steepest descent on the potential field, a succession of points with lower potential values leading to the point with least potential (goal) is found out. It is observed by Connolly et al [4] that this process guarantees a path to the goal without encountering local minima and successfully avoiding any obstacle, as a harmonic function cannot possess an extremum value except at the domain boundary.…”

Section: Introductionmentioning

confidence: 99%

A Comparative Study of Dirichlet and Neumann Conditions for Path Planning through Harmonic Functions

Karnik

DasGupta

Eswaran

2002

Lecture Notes in Computer Science

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Abstract. Harmonic functions, by virtue of their extrema appearing only on the domain boundary, are known to have an advantage as a global potential function in the potential field based approach for robot path planning. However, a wide range of possibilities exist for the global harmonic function for a particular situation, depending on the boundary conditions. This paper conducts a comparison of two major classes of boundary conditions, namely Dirichlet and Neumann conditions, and attempts to discover their relative merits and demerits. It is found that the Neumann conditions offer a surer and faster approach to the path planning problem, though suffering from the disadvantage of occasional tendency of the planned path to graze along the domain boundary.

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Path planning using Laplace's equation

Cited by 389 publications

References 5 publications

Planning active cannula configurations through tubular anatomy

Planning active cannula configurations through tubular anatomy

Solving the potential field local minimum problem using internal agent states

A Comparative Study of Dirichlet and Neumann Conditions for Path Planning through Harmonic Functions

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