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This tutorial covers the use of harmonic functions to navigate a robot through an obstacle course. Harmonic potential fields overcomes many of the shortcomings of ordinary potential fields path planning algorithms, as it utilizes harmonic functions that rid of local minima within a plane.

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This tutorial's motivation is to first give an overview of properties of harmonic functions, and then demonstrate the ability of harmonic functions to alleviate some of the shortcomings typical potential fields path planning algorithms face. I will then explain and demonstrate the multi-panel approach for complex obstacle representation.

The rest of this tutorial is presented as follows:

• Harmonic Functions
• Properties of Harmonic Functions
• Uniform Flow
• Panel Method for Potential Fields
• Multi-Panel Method for Complex Obstacles
• Final Words

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• Harmonic functions are multi-variable functions defined in terms of the laplacian.
• A laplacian is a special way to extend the second-derivative into multiple dimensions.
• Harmonic functions are functions where the laplacian is equal to zero (at every possible input point ε, around a central point μ, with radius r).

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• Mean-Value Property

• For a 2-dim potential function φ, the mean-value property states that value of harmonic function φ at a point (x,y) is equal to average value at φ over any circle around (x,y)

• Maximum Principle

• Maximum of a non-constant harmonic function occurs on the boundary

• Minimum Principle

• Minimum of a non-constant harmonic function occurs on the boundary

• These properties are useful for obstacle avoidance as harmonic function completely eliminates local minima, a major shortcoming of conventional potential field path planning algorithms.

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• Harmonic Function useful for building artificial potential fields.
• In n=2 dimensions, if flow flows in directions which makes an angle α with x-axis, the potential function for uniform flow is:
• 
• U = magnitude (strength of uniform flow)
• With uniform flow, potential around a point is determined by strength of: uniform flow + strength of panel source.
• In the above equation φ,

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• The panel method is used to solve potential flow of a fluid around bodies of arbitrary shape.
• In this method, the surface of a body is covered by a finite number of small areas called panels.
• Each panel contains source or sink singularities which have a uniform density.
• In the picture below, a single panel is distributed with uniform sources with strength per unit length of λ.
• 

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• Minimum Principle

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