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--- harmonic_potential_fields [2017/09/12 17:52] – tbrodeur
+++ harmonic_potential_fields [2017/09/14 12:58] (current) – tbrodeur
@@ Line 36: / Line 36: @@
   * [[Harmonic_Potential_Fields#Panel Method for Potential Fields | Panel Method for Potential Fields]]
   * [[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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 ===== Multi-Panel Method for Complex Obstacles =====
+  * {{: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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