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--- harmonic_potential_fields [2017/09/12 14:49] – tbrodeur
+++ harmonic_potential_fields [2017/09/14 12:58] (current) – tbrodeur
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-<fc #008000><fs x-large> Harmonic Functions for Potential Field Navigation </fs></fc>
+<fc #008000><fs x-large> Potential Field Navigation with Harmonic Functions</fs></fc>
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@@ Line 31: / Line 31: @@
 The rest of this tutorial is presented as follows:
-  * [[potential_fields#Properties of Harmonic Functions| Properties of Harmonic Functions]]
+  * [[Harmonic_Potential_Fields#Harmonic Functions | Harmonic Functions]]
-  * [[potential_fields#Harmonic Functions | Harmonic Functions]]
+  * [[Harmonic_Potential_Fields#Properties of Harmonic Functions| Properties of Harmonic Functions]]
-  * [[potential_fields#Uniform Flow | Uniform Flow]]
+  * [[Harmonic_Potential_Fields#Uniform Flow | Uniform Flow]]
-  * [[potential_fields#Panel Method for Potential Fields | Panel Method for Potential Fields]]
+  * [[Harmonic_Potential_Fields#Panel Method for Potential Fields | Panel Method for Potential Fields]]
-  * [[potential_fields#Multi-Panel Method for Complex Obstacles | Multi-Panel Method for Complex Obstacles]]
+  * [[Harmonic_Potential_Fields#Multi-Panel Method for Complex Obstacles | Multi-Panel Method for Complex Obstacles]]
-  * Final Words
+  * [[Harmonic_Potential_Fields#Code | Code]]
+  * [[Harmonic_Potential_Fields#Final Words | Final Words]]
+This work is presented in more detail in [[https://pdfs.semanticscholar.org/6715/e34f4e233ec7ca4c8e88f5b5d282e5bafc20.pdf|this paper ]].
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-===== Properties of Harmonic Functions =====
+===== Harmonic Functions =====
-  * **<fs medium> Minimum Principle </fs>**
+  * 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 (Δf = 0).
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-===== Harmonic Functions =====
+===== Properties of Harmonic Functions =====
+  * **<fs medium> Mean-Value Property </fs>**
+         * 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)
+  * **<fs medium> Maximum Principle </fs>**
+       *Maximum of a non-constant harmonic function occurs on the boundary
   * **<fs medium> Minimum Principle </fs>**
+       *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.
+    * {{:screen_shot_2017-09-12_at_4.40.20_pm.png?400|}}
+    * The figure above and to the left represents an artificial potential field using harmonic functions, whereas the right represents that of an artificial potential field using non-harmonic function. There exists a local minima at (0,0) for the non-harmonic function but not so for the harmonic.
+    * You can also observe the minimum and maximum principles displayed in the figure. All maximum and minimums occur on the boundary of the potential field.
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 ===== Uniform Flow =====
-  * **<fs medium> Minimum Principle </fs>**
+  * 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:
+  * {{:screen_shot_2017-09-12_at_4.05.16_pm.png?200|}}
+  * U = magnitude (strength of uniform flow)
+  * With uniform flow, potential around a point is determined by strength of: uniform flow + strength of [[Harmonic_Potential_Fields#Panel Method for Potential Fields | panel source]].
+  * In the above equation φ, {{:screen_shot_2017-09-12_at_4.00.31_pm.png?200|}}
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@@ Line 64: / Line 91: @@
 ===== Panel Method for Potential Fields =====
-  * **<fs medium> Minimum Principle </fs>**
+  * 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.
+  * {{:screen_shot_2017-09-12_at_4.11.04_pm.png?500|}}
+  * The potential at any point (x,y) induced by the sources contained with a small element dl of the panel is: {{:screen_shot_2017-09-12_at_4.55.22_pm.png?300|}}
+  * To find the induced potential function of the whole panel, take the integral over the length of the panel: {{:screen_shot_2017-09-12_at_4.55.26_pm.png?300|}}
+  * Differentiation w.r.t. x and y gives the following expressions for the velocity components:
+       {{:screen_shot_2017-09-12_at_4.55.31_pm.png?300|}}{{:screen_shot_2017-09-12_at_4.55.37_pm.png?300|}}
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 ===== Multi-Panel Method for Complex Obstacles =====
-  * **<fs medium> Minimum Principle </fs>**
+  * {{:screen_shot_2017-09-14_at_11.21.22_am.png?500|}}
+  * Used to represent complex obstacles and cluttered environments.
+  * Obstacles approximated by set of panels numbered in clockwise direction.
+  * Each panel has own center point with a desired outward normal velocity as input variable.
+  * Boundary points are intersections of neighboring panels.
+  * If we let M = # of panels, and let λ<sub>1</sub>, λ<sub>2</sub>, ... λ<sub>M</sub> represent the source/sink strength per unit length of panel M, then the velocity potential at any point (x,y) by panel j is: {{:screen_shot_2017-09-14_at_11.14.37_am.png?300|}}
+  * where R<sub>j</sub> is the euclidean distance between point (x,y) and the point (x<sub>j</sub>, y<sub>j</sub>) at panel j.
+== Goal Points ==
+  * We need an attractive potential at our goal point, where the potential has only one global minimum.
+  * This potential can be represented by a point singularity of sink, that acts like a drain in a sink,  and has a strength of A > 0, and can be represented by: {{:screen_shot_2017-09-14_at_11.35.53_am.png?200|}}
+  * where R<sub>g</sub> is the euclidean distance between point (x,y) and the goal point (x<sub>g</sub>, y<sub>g</sub>).
+== Potential Functions ==
+  * The total potential due to obstacles, goal, and uniform flow is:
+    {{:screen_shot_2017-09-14_at_12.34.57_pm.png?350|}} {{:screen_shot_2017-09-14_at_12.35.02_pm.png?500|}}
+  *
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